Method of writing a composite 1-step hologram

ABSTRACT

A method of writing a single or double parallax composite 1-step hologram is disclosed. Digital data is acquired from a real or virtual object and is described by a luminous intensity tensor. A single mathematical transformation is performed to convert the luminous intensity tensor into a mask tensor. The single mathematical transformation transforms the digital data whilst integrally correcting the digital data for the finite distortion of an optical objective. Corrected data, described by the mask tensor, is written on to a spatial light modulator. A laser beam is directed on to the spatial light modulator so that at least a portion of the laser beam is spatially modulated to form a spatially modulated laser beam. The spatially modulated laser beam is then passed through an optical objective having a finite distortion and in combination with a reference recording beam forms a composite hologram.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to methods of writing composite 1-stepholograms.

2. Description of Related Art

PCT applications WO01/45943 and WO01/42861 (Brotherton-Ratcliffe et al.)described a holographic printing system based on a rapid pulsed laser.This system was able to operate in two modes (“a dual-mode printer”). Inthe first mode, a holographic master hologram, also known as an H1, isproduced. This hologram is then copied conventionally to a second andfinal hologram, known as the H2, by machines similar to those describedby Grichine et al. (1997). In the second mode of operation, the finalhologram is written directly.

In its first mode of operation, the digital data required by the systemdisclosed in WO01/45943 and WO01/42861 consists of ordinary perspectiveviews that are easily generated by standard 3D commercial softwarepackages. These images are then trivially distorted in order tocompensate for inherent optical distortion that is present in theinvention.

In its second mode of operation, the digital data required by the systemdisclosed in WO01/45943 and WO00/42861 are derived by applying variousgeneral mathematical transformations on the ensemble of the undistorteddata-set that would be used for the generation of a hologram under thefirst mode of operation.

U.S. Pat. No. 5,793,503 (Haines et al.) described varioustransformations for the preparation of digital data for writing thehologram of a 3D computerized model using an ideal hologram printer.This arrangement concentrated on the treatment of data from aspecialized 3D model but also treated the case of the generation of therequired digital data from the conventional perspective views producedby standard commercial software.

The methods described in U.S. Pat. No. 5,793,503 are inappropriate forapplication to the type of holographic printers described in PCTapplications WO01/45943 and WO01/42861. One reason for this is that suchprinters intrinsically possess very large optical distortion caused by afinite 5^(th) order Seidel coefficient in the writing objective. Thisdistortion, which generally varies from pixel to pixel in the case ofnon-static SLM printers should not be corrected for independently bysequential application of correction algorithms as this would lead toboth image noise and computational disadvantage.

Another reason why the arrangement disclosed in U.S. Pat. No. 5,793,503is inappropriate in the present context is that the 1-step hologramsthat are produced by the system disclosed in PCT applications WO01/45943and WO01/42861 must, on display, generally be illuminated by a lightfrom a point source which does not correspond geometrically to therecording illumination used within the printer and thus any propermethod, for transforming perspective data into the required data, shouldbe based on a general diffraction model. In its generality such a modelmust take into account such parameters as the emulsion and hologramsubstrate refractive indices as well as the recording and ray replaygeometry.

Although the system disclosed in WO01/45943 and WO01/42861 represents aconsiderable advance over the prior-art it too suffers from variouslimitations. In particular, by not integrating the correction foroptical distortion of the objective, into the data rearrangementtransforms necessary for writing 1-step holograms, image quality isinevitably compromised. In addition, by only tracking the referencerecording beam in one dimension, rather than two, the printer isfundamentally unable to produce large format holograms that areilluminated by close point-source lights. Finally, by seeking to correctfor geometrical image distortions alone the prior-art printer suffersfrom increasing discoloration effects as a closer (and more realistic)point-source illumination is demanded.

SUMMARY OF THE INVENTION

According to an aspect of the present invention there is provided amethod of writing a composite 1-step hologram, comprising:

generating a laser beam;

acquiring digital data from an object, the digital data being describedby a luminous intensity tensor ^(kg)I_(ij) wherein i and j are thehorizontal and vertical pixel coordinates of a given perspective viewthat is generated by a real or virtual camera whose location isdescribed by k in the horizontal dimension and g in the verticaldimension;

performing a single mathematical transformation to convert the luminousintensity tensor ^(kg)I_(ij) into a tensor ^(μν)T_(αβ) wherein α and βare the horizontal and vertical coordinates of a holographic pixel onthe composite hologram and μ and ν are the horizontal and verticalcoordinates of a given pixel on a spatial light modulator on to whichthe data for each holographic pixel is written, wherein the singlemathematical transformation transforms the digital data whilstintegrally correcting the digital data for the finite distortion of anoptical objective;

writing corrected data to a spatial light modulator, wherein thecorrected data is described by the tensor ^(μν)T_(αβ);

directing the laser beam on to the spatial light modulator so that atleast a portion of the laser beam is spatially modulated by the spatiallight modulator to form a spatially modulated laser beam;

passing the spatially modulated laser beam through an optical objectivehaving a finite distortion, the optical objective focusing the spatiallymodulated laser beam on to a photosensitive is substrate;

directing a reference recording beam on to the photosensitive substrate;and

forming a double-parallax composite hologram on the photosensitivesubstrate.

According to another aspect of the present invention, there is provideda method of writing a composite 1-step hologram, comprising:

generating a laser beam;

acquiring digital data from an object, the digital data being describedby a luminous intensity tensor ^(k)I_(ij) wherein i and j are thehorizontal and vertical pixel coordinates of a given perspective viewthat is generated by a real or virtual camera whose location isdescribed by k in the horizontal dimension;

performing a single mathematical transformation to convert the luminousintensity tensor ^(k)I_(ij) into a tensor ^(μν)T^(αβ) wherein α and βare the horizontal and vertical coordinates of a holographic pixel onthe composite hologram and μ and ν are the horizontal and verticalcoordinates of a given pixel on a spatial light modulator on to whichthe data for each holographic pixel is written, wherein the singlemathematical transformation transforms the digital data whilstintegrally correcting the digital data for the finite distortion of anoptical objective;

writing corrected data to a spatial light modulator, wherein thecorrected data is described by the tensor ^(μν)T_(αβ);

directing the laser beam on to the spatial light modulator so that atleast a portion of the laser beam is spatially modulated by the spatiallight modulator to form a spatially modulated laser beam;

passing the spatially modulated laser beam through an optical objectivehaving a finite distortion, the optical objective focusing the spatiallymodulated laser beam on to a photosensitive substrate;

directing a reference recording beam on to the photosensitive substrate;and

forming a single-parallax composite hologram on the photosensitivesubstrate.

The digital data may be acquired from a real object and comprises aplurality of perspective views of the required hologram image.Alternatively, the digital data may be acquired from a virtual objectand comprises a plurality of perspective views of the required hologramimage.

The composite 1-step hologram may comprise a transmission hologram or areflection hologram.

According to one embodiment, the single mathematical transformationgenerates a rectangular viewing window located in front of the hologram.The spatial light modulator may either be static whilst writing thehologram or the spatial light modulator may be moving whilst writing thehologram. The viewing window is preferably either of substantiallysimilar size to the composite hologram or of different size to thecomposite hologram. The viewing window may be symmetrically located infront of the composite hologram or generally offset from the centre ofthe composite hologram. Preferably, either the viewing window is locatedat the same perpendicular distance from a given point in the holographicimage as the camera plane is located from the corresponding point on theobject from which the digital data is acquired or the viewing window islocated at a certain perpendicular distance from a given point in theholographic image and the camera plane is located at a substantiallydifferent perpendicular distance from the corresponding point on theobject from which the digital data is acquired.

The digital data may be generated by a real or virtual camera whichgenerates either a plurality of apodized images which are centred in aframe which corresponds with the object which is to be reproduced by thehologram or a plurality of non-apodized images having frames whichcorrespond with the object which is to be reproduced by the hologram,the frames being generally off-centred.

According to an embodiment the single mathematical transformationgenerates a scrolling viewing window located in front of the hologram.

According to another embodiment the single mathematical transformationgenerates a viewing window having a fixed size in the horizontaldimension and which scrolls in the vertical dimension, the viewingwindow being located in front of the hologram.

According to another embodiment the single mathematical transformationgenerates a viewing window having a fixed size in the vertical dimensionand which scrolls in the horizontal dimension, the viewing window beinglocated in front of the hologram.

Preferably either the camera plane is located at a certain distance froma point on the object and the viewing plane is located at substantiallythe same distance from a corresponding point in the holographic image orthe camera plane is located at a certain distance from a point on theobject and the viewing plane is located at substantially a differentdistance from a corresponding point in the holographic image.

According to one embodiment the composite hologram is formed using avariable angle reference recording beam. According to another embodimentthe composite hologram is formed using a fixed angle or collimatedreference recording beam. The composite 1-step hologram may be replayedusing a point-source light or alternatively using collimated light.

Preferably, the single mathematical transformation additionallyintegrally corrects the digital data for the image distortion caused bythe altitudinal and azimuthal reference beam angle(s) used to replayeach holographic pixel of the hologram being different from thealtitudinal and azimuthal reference beam angle(s) used to write eachholographic pixel of the hologram.

Preferably, the single mathematical transformation additionallyintegrally pre-distorts the digital data so that the data written on tothe spatial light modulator is distorted. The reference recording beamis preferably overcorrected using either an astigmatic or anon-astigmatic geometry.

Preferably, individual altitudinal and azimuthal reference recordingangles are determined for at least a majority, preferably all, of theholographic pixels forming the hologram.

According to an embodiment the overlap of viewing windows of a pluralityof holographic pixels is arranged to be maximised. The overlap ofviewing windows of two diagonally opposed holographic pixels may bemaximised. Preferably, either the angular resolution within the overallviewing window of the hologram averaged over the overall viewing windowis maximised or the angular resolution at the periphery of the overallviewing window is maximised. The pre-distortion of the digital data andthe step of overcorrecting the reference recording beam are preferablyarranged such that chromatic discoloration is minimized.

According to an embodiment the single mathematical transformationadditionally integrally corrects the digital data for the distortioncaused by emulsion swelling of the substrate.

According to an embodiment the single mathematical transformationadditionally integrally corrects the digital data for the distortioncaused by the wavelength of light used to replay the hologram beingdifferent from the wavelength of light used to write the hologram.

According to a particularly preferred embodiment, a plurality of colourchannels are provided. According to an embodiment a red and/or greenand/or blue colour channel are provided. A spatial light modulator maybe provided for each colour channel. The composite 1-step hologrampreferably comprises a multiple colour hologram. The multiple colourhologram is formed using reference recording beams having a firstgeometry and the hologram is replayed with light rays having a geometrydifferent to the first geometry.

The actual replay wavelength may be calculated as a function ofaltitudinal and azimuthal angles for at least a majority, preferablyall, of the holographic pixels forming the hologram.

Linear chromatic coupling tensors for each colour channel may becalculated. A separate tensor ^(μν)T^(αβ) is preferably calculated foreach primary colour channel. Preferably, a corrected tensor ^(μν)T_(αβ)is then calculated for each primary colour channel as a linearcombination of the uncorrected component colour tensors ^(μν)T_(αβ) eachoperated on by respective chromatic coupling tensors. For eachholographic pixel each of the corrected tensors is preferably written toa separate spatial light modulator in such a way as to create a fullycolour-corrected composite colour hologram.

For a double parallax case, the single mathematical transformationbetween the tensors ^(kg)I_(ij) and ^(μν)T_(αβ) preferably consists of areordering of the elements according to a set of single index laws ofthe form k=f₁(α,β,μ,ν,P_(l),Q_(l),H_(l),λ),g=f₂(α,β,μ,ν,P_(l),Q_(l),H_(l),λ), i=f₃(α,β,μ,ν,P_(l),Q_(l), H_(l),λ)and is j=f₄(α,β,μ,ν,P_(l),Q_(l),H_(l),λ), wherein the functions f_(n)are general functions of the indicated indices, P_(l) are a set ofparameters characterizing the physical characteristics of the hologram,Q_(l) are a set of parameters characterizing the optical properties ofthe hologram writing mechanism, H_(l) is a set of parameterscharacterizing the geometrical properties of the reference recording andreference replay beams and λ is the wavelength at which the hologram isrecorded.

For a single parallax case, the single mathematical transformationbetween the tensors ^(k)I_(ij), and ^(μν)T_(αβ) preferably consists of areordering of the elements according to a set of single index laws ofthe form k=f₁(α,β,μ,ν,P_(l),Q_(l),H_(l),λ), i=f₂(α,β,μ,ν,P_(l),Q_(l),H_(l),λ) and j=f₃(α,β,μ,ν,P_(l),Q_(l),H_(l),λ), wherein the functionsf_(n) are general functions of the indicated indices, P_(l) are a set ofparameters characterizing the physical characteristics of the hologram,Q_(l) are a set of parameters characterizing the optical properties ofthe hologram writing mechanism, H_(l) is a set of parameterscharacterizing the geometrical properties of the reference recording andreference replay beams and λ is the wavelength at which the hologram isrecorded.

According to another aspect of the present invention there is provided a1-step holographic printer, comprising:

a laser source;

control means for acquiring digital data from an object, the digitaldata being described by a luminous intensity tensor ^(kg)I_(ij); whereini and j are the horizontal and vertical pixel coordinates of a givenperspective view that is generated by a real or virtual camera whoselocation is described by k in the horizontal dimension and g in thevertical dimension, the control means performing a single mathematicaltransformation to convert the luminous intensity tensor ^(kg)I_(ij) intoa tensor ^(μν)T_(αβ) wherein α and β are the horizontal and verticalcoordinates of a holographic pixel on the composite hologram and μ and νare the horizontal and vertical coordinates of a given pixel on aspatial light modulator is on to which the data for each holographicpixel is written, wherein the single mathematical transformationtransforms the digital data whilst integrally correcting the digitaldata for the finite objective distortion of an optical objective;

a spatial light modulator onto which data described by the tensor^(μν)T_(αβ) is written in use, wherein in use a laser beam is directedon to the spatial light modulator so that at least a portion of the beamprofile of the laser beam is spatially modulated by the spatial lightmodulator to form a spatially modulated laser beam; and

an optical objective through which the spatially modulated laser beam ispassed in use, the optical objective focusing in use the spatiallymodulated laser beam on to a photosensitive substrate so that adouble-parallax composite hologram is formed in use on to aphotosensitive substrate.

According to another aspect of the present invention there is provided a1-step holographic printer, comprising:

a laser source;

control means for acquiring digital data from an object, the digitaldata being described by a luminous intensity tensor ^(k)I_(ij) wherein iand j are the horizontal and vertical pixel coordinates of a givenperspective view that is generated by a real or virtual camera whoselocation is described by k in the horizontal dimension, the controlmeans performing a single mathematical transformation to convert theluminous intensity tensor ^(k)I_(ij) into a tensor ^(μν)T_(αβ) wherein αand β are the horizontal and vertical coordinates of a holographic pixelon the composite hologram and μ and ν are the horizontal and verticalcoordinates of a given pixel on a spatial light modulator on to whichthe data for each holographic pixel is written, wherein the singlemathematical transformation transforms the digital data whilstintegrally correcting the digital data for the finite objectivedistortion of an optical objective;

a spatial light modulator onto which data described by the tensor^(μν)T_(αβ) is written in use, wherein in use a laser beam is directedon to the spatial light modulator so that at least a portion of the beamprofile of the laser beam is spatially modulated by the spatial lightmodulator to form a spatially modulated laser beam; and

an optical objective through which the spatially modulated laser beam ispassed in use, the optical objective focusing in use the spatiallymodulated laser beam on to a photosensitive substrate so that asingle-parallax composite hologram is formed in use on to aphotosensitive substrate.

The preferred embodiment provides a class of methods that is capable ofeffectively and efficiently transforming 2D digital perspective viewsthat have been derived from a standard 3D computer model into data whichis then used by a holographic printer to produce a 1-step compositehologram. Unlike the arrangement disclosed in U.S. Pat. No. 5,793,503which is primarily directed towards the generation of small holograms in1-step printers, the preferred embodiment is directed primarily towardsthe generation of larger holograms using either 1-step or dual-modeprinters.

According to the preferred embodiment, rainbow or reflection typeholograms may be produced. Rainbow holograms of a single colour channelor many colour channels may be generated. Equally reflection hologramsof one or more colours are provided for, in addition to achromaticholograms of both rainbow and transmission nature. Particular attentionis paid to reflection holograms (both single and multiple-colour andboth single and double parallax).

The SLM may be moved which is preferred for printers that print both1-step and 2-step holograms (dual mode printers).

According to a preferred embodiment holograms may be written which havedifferent replay and recording reference geometries.

According to another embodiment the image data is transformed and therecording reference beam within the printer is adjusted preferablytwo-dimensionally in order to optimize the final viewing window of thehologram and to produce a distortion-free image, given the intendeddisplay illumination.

According to another embodiment the image data is transformed in orderto produce a distortion free image and the recording reference beamwithin the printer is adjusted preferably two-dimensionally in order tooptimize the final viewing window of the hologram and to maximizeangular image resolution, given the intended display illumination.

According to another embodiment the image data is transformed in orderto produce a distortion free image and the recording reference beamwithin the printer is adjusted preferably two-dimensionally in order tooptimize the final viewing window of the hologram, to maximize angularimage resolution and/or to minimize chromatic discoloration, given theintended display illumination.

According to another embodiment only one single set of numerical indexrules is formulated and applied to convert raw image data into datarequired by the printer SLM(s), the set integrating the basic datareordering required for 1-step holograms with correction for each of avariety of geometrically distorting effects.

According to another embodiment the intrinsic chromatic discoloration ofa 1-step hologram, arising when the recording and replay geometriesdiffer, is corrected for by numerical transformation of the image data.

BRIEF DESCRIPTION OF THE DRAWINGS

Various embodiments of the present invention together with otherarrangements given for illustrative purposes only will now be described,by way of example only, and with reference to the accompanying drawingsin which:

FIG. 1 illustrates a plan view of a known holographic printer;

FIG. 2 illustrates the known holographic printer working in an H1 masterwriting mode for the case of a transmission H1 hologram;

FIG. 3 illustrates the known holographic printer working in an H1 masterwriting mode for the case that the holographic recording material isorientated at the achromatic angle;

FIG. 4 illustrates the known holographic printer working in an H1 masterwriting mode for the case of a reflection H1 hologram;

FIG. 5 illustrates the known holographic printer working in direct(1-step) writing mode for the case of a reflection hologram;

FIG. 6( a) illustrates the overlapping object beam density patternrecorded on the holographic material typical of an H1 master hologramwritten for the creation of a rainbow hologram by conventional transferwith each circle containing the perspective information for a certainviewpoint;

FIG. 6( b) illustrates the overlapping object beam density patternrecorded on the holographic material typical of an H1 master hologramwritten for the creation of a full-colour rainbow hologram byconventional transfer with each ellipse containing the perspectiveinformation for a certain viewpoint, the three rows representing thethree primary colour separations;

FIG. 7 illustrates the overlapping object beam density pattern recordedon the holographic material typical of an H1 full aperture masterhologram written for the creation of a mono or full colour reflectionhologram by conventional transfer with each circle containing theperspective information from a certain point in space as shown in FIG.9;

FIG. 8 illustrates the object beam density pattern recorded on theholographic material typical of a directly written (1-step) hologramwith each circle containing the directional and amplitude information oflight originating from that point that constitutes the 3-D image;

FIG. 9 illustrates the process of acquiring data from a series ofsequential camera shots that can be used to generate the holograms—thediagram is also used to represent a computer model of an object where aviewing (or camera) plane is defined on which perspective views aregenerated;

FIG. 10 illustrates the recording of various perspective views of anobject or virtual object and the mathematical discretization of thecamera tracking plane;

FIG. 11 illustrates the mathematical discretization of each camera shot;

FIG. 12 illustrates the mathematical discretization of the hologram;

FIG. 13 illustrates the simplified recording scheme (seen from top) bywhich a hologram is written;

FIG. 14 a illustrates the recording geometry (single parallax) for atranslating camera that is always pointing forwards and of fixed fieldof view (FOV);

FIG. 14 b illustrates the hologram geometry (seen from top) for a fixedviewing window that is equal in shape and size to the actual hologram(simple translating camera, fixed SLM, single parallax);

FIG. 15 illustrates the centred camera geometry seen from top);

FIG. 16 illustrates a centred camera geometry (seen from top) for thecase of a hologram viewing window equal in size and shape to thehologram (single parallax, fixed SLM);

FIG. 17 illustrates the centering of the camera for the maximum FOV case(seen from top);

FIG. 18 illustrates the hologram geometry (seen from top) for themaximum FOV centred camera case (single parallax, fixed SLM);

FIG. 19 illustrates the geometry (seen from the side) for a restrictedvertical viewing window of height Γ (the hologram is to the left and theviewing zone to the right);

FIG. 20 illustrates the geometry for a general rectangular window withoversized SLM (seen from top);

FIG. 21 illustrates the geometry for a general rectangular viewingwindow with tracking SIM and centred camera (seen from top);

FIG. 21 a illustrates, for the case of a tracking SLM, the relationshipbetween the horizontal position of the centre of the SLM relative to thecentre of the objective and the horizontal position of the centre of theprojected image of the SLM at the viewing plane, again relative to thecentre of the objective;

FIG. 22 illustrates the geometry for a general rectangular window withtracking SLM and centred camera (seen from the side);

FIG. 23 illustrates the geometry for the full-parallax case withtranslating camera (seen from the side);

FIG. 24 illustrates the geometry for the full-parallax case with acentred camera (seen from the side);

FIG. 25 illustrates the geometry for the full-parallax case, centredcamera and maximum FOV scrolling windows (side view);

FIG. 25 a illustrates an offset geometry whereby the viewing window isoffset from the centre of the hologram—case of a general rectangularwindow—seen from the top;

FIG. 25 b illustrates an offset geometry whereby the viewing window isoffset from the centre of the hologram—case of a general rectangularwindow & single horizontal parallax—seen from the side;

FIG. 26 illustrates the treatment of objective distortion: NormalizedObject (left) and Image (right) planes;

FIG. 27 illustrates the case of a moving SLM;

FIG. 28 illustrates the right-handed spherical coordinate system usedfor the diffraction analysis—(a) recording geometry and (b) playbackgeometry;

FIG. 29 a illustrates the geometry for a point-source reference;

FIG. 29 b illustrates the geometry for a point-source reconstruction;

FIG. 29 c illustrates the geometry for the object point on recording;

FIG. 29 d illustrates the geometry for the image point onreconstruction;

FIG. 29 e illustrates the geometry for a point sink reference;

FIG. 30 illustrates a numerical simulation of the viewing windows of a1-step monochromatic reflection hologram recorded at 526.5 nm, 80 cm×60cm, viewed from the front, written using a collimated reference beam at56 degrees angle of incidence and illuminated by a point source locatedat a 3 m distance and at a 56 degrees axial angle of incidence(refractive index of the emulsion is 1.63);

FIG. 31 illustrates a numerical simulation of a 2 m×2 m monochromaticreflection hologram, recorded under a collimated reference geometry andilluminated from a distance of 3 m by a point source (viewing distanceis 2 m, reference recording and axial replay angles are 56 degrees tothe normal—otherwise all other parameters as in FIG. 30);

FIG. 32 a illustrates the case of FIG. 30 but with an exactly conjugatereference beam at recording giving undistorted viewing windows on replayby a point source;

FIG. 32 b illustrates the case of FIG. 30 but with an overcompensatedconverging reference beam showing the merging of the two diagonallyopposed viewing zones (Labels as in FIG. 30);

FIG. 32 c illustrates the case of FIG. 32( b) but using an astigmaticconverging reference beam to bring the windows into vertical as well ashorizontal alignment(Labels as in FIG. 30);

FIG. 33 illustrates the calculation of the P function for the case oftwo diagonally separated pixels where the path variable used in theintegration is either Λ* or Λ.

DESCRIPTION OF THE INVENTION

WO01/45943 and WO01/42861 disclose a dual-mode holographic printer basedon pulsed laser technology and WO01/29487 and WO02/29500 disclose amultiple-colour pulsed laser which may be employed in the holographicprinter. The known holographic printer is capable of producing eitherfinal 1-step holograms or H1 master holograms for H2 transfer. Hologramscan be either of the reflection or transmission type. They may have fullor limited parallax. They may be monochrome, rainbow or full-colour. Theprinting speed of the printer is several orders of magnitude greaterthan other known holographic printers. In addition it is compact andhologram quality is independent of external environmental noise.

In order to understand the application of the present invention thesalient features of the known holographic printer will now be reviewed.For simplicity and clarity the case of a single monochromatic laser willbe considered. FIG. 1 shows an overhead view of the known holographicprinter. A single colour single-frequency pulsed laser 100 capable ofrapid operation (typically 20 Hz) and having sufficient temporalcoherence emits a beam of coherent light which is split by a variablebeamsplitter 101. The beam 102 continues to the mirror 103 whereupon itis diverted to the mirror 104 whereupon it is diverted to the waveplate105 which controls the polarization of the beam. The beam continues to atelescope comprising lenses 106, 107 and 167. Lens 107 is mounted on amotorized translation stage 108 with motor 109. The diameter of the beamexiting from optic 107 is thus controlled and approximately collimated.The beam passes to the micro-lens array 110 which expands it onto thecollimating lens assembly 111. The distance between the elements 110 and111 is chosen to be the effective focal length of the lens 111. In sucha way a “collimated” beam exits the optic 111 with a controllablespatial coherence. The beam now illuminates a liquid crystal display(SLM—spatial light modulator) 112, having resolution 768×1024 pixels andlateral dimension of 26.4 mm, which is mounted on a 2-D motorizedtranslation stage 116 having vertical control motor 115 and horizontalcontrol motor 118. Positions of maximum SLM horizontal displacement areindicated by 113 and 114. The SLM position is adjusted when writing H1type holograms and is used to attain a much higher resolution of finalimage than would otherwise be possible with the same static SLM for agiven angle of view. The SLM position may also be adjusted when writinga 1-step hologram in order to maintain a particular hologram viewingwindow geometry.

After passing through the liquid crystal display, the beam traverses alinear polarizer that converts the SLM image from a polarizationrotation image into amplitude modulation. Then the beam passes throughthe wide-angle objective 119 mounted on the motorized translation stage120 with motor 163. This stage is used to control the position of thefocused image of the SLM produced by the objective 119. The size of theminimum waist 166 of the object beam is controlled by the motorizedstage 108 with motor 109. The object beam now comes to bear on thephotosensitive material 162 here shown as film mounted on a roll/stagesystem. The motor 129 controls movement of the stage 123 towards andaway from the position of minimum object beam waist. The rollers 124 and125 control the horizontal movement of the film 162 in front of theobject beam. The motor 128 controls the vertical movement of the film infront of the object beam. Motor 126 controls the motion of the rollers124 and 125. Rollers 122 and 131 tension the film and control thehorizontal angle that the film makes to the axial propagation vector ofthe object beam.

The reference beam is split from the main laser beam by the variablebeamsplitter 101 controlled by motor 165. The beam 135 is directed to amirror 136 whereupon it is reflected through an quasi-elliptical orrectangular aperture 137, an effective image of which is eventuallycreated at the intersection of the reference beam with the holographicrecording material, such quasi-elliptical or rectangular shape producinga defined circular or quasi-elliptical or rectangular referencefootprint on the recording material as may be required by the type ofhologram being written. The reference beam continues to the waveplate138 which controls the polarization of the laser beam. The elements 139and 141 with either 164 or 163 form a telescope that controls the sizeof the beam after 164/163 which is adjustable by the motorized stage 142with motor 143. The beamsplitter switch 144 either directs the referencebeam on the path 154 or onto the path 145. Path 145 is used to createtransmission holograms whereas path 154 is used to create reflectionholograms.

In the case of path 145 the reference beam passes through the lens 164that produces an approximate image of the aperture 137 at the recordingmaterial surface. This lens also corrects for the slight divergence ofthe light produced by the lens 141. The divergence of the light after164, which is ideally collimated, is thus controlled to withindiffraction limits. Practically this means that for small reference beamsize the beam will not be exactly collimated but that such departurefrom collimation will lead to an image blurring significantly less thanthat induced by the source size of the final hologram illuminationsource. Mirrors 146 and 149 now direct the reference beam onto itstarget to intersect the object beam at the surface of the holographicrecording material. Motorized rotation stages 147 and 150 with motors148 and 152 respectively and the linear translation stage 151 with motor153 assure that different reference angles may be achieved for differentplacements and orientations of the recording material.

In the case of path 154 the reference beam passes through the lens 163that produces an approximate image of the aperture 137 at the recordingmaterial surface. This lens also corrects for the slight divergence ofthe light produced by the lens 141. The divergence of the light after163, which is ideally collimated, is thus controlled to withindiffraction limits as above. Mirrors 155 and 156 now direct thereference beam onto its target to intersect the object beam at thesurface of the holographic recording material, this time from theopposite side to the object beam. The motorized rotation stage withmotor 159 and the linear translation stage 158 with motor 160 assurethat different reference angles may be achieved for different placementsand orientations of the recording material.

The known holographic printer can function in a variety of differentmodes. FIG. 2 shows a diagram of the system in H1 transmission mode.Note that the reference beam comes in towards the recording materialfrom the same side as the object beam to form a holographic pixel 121.Note also that this pixel is significantly displaced from the point ofminimum waist 166. The image of the SLM 112 is located at a distance 201from the recording material 162 and as such a screen placed at 202 wouldshow a sharply focused image of each 2-D picture loaded into the SLM112. The plane 202 usually corresponds to the H2 plane in a transfergeometry.

In order to record an H1 transmission hologram perspective views of areal or computer generated object are loaded into the SLM one by one, aholographic pixel recorded, the recording material advanced and theprocess repeated for each image. For the case of the generation of arainbow transmission master hologram a line of pixels is written ontothe holographic recording material as illustrated in FIG. 6( a). Eachcircle represents an interference pattern containing information about acertain perspective view along a horizontal viewing line. FIG. 6( b)illustrates the case pertaining to the generation of an RGB rainbowhologram master where three lines of pixels are written at theachromatic angle each line corresponding to a red, green or bluecomponent image in the axial viewing position of the final hologram. Therecording geometry for FIG. 6( b) is shown in FIG. 3. In order to recordan H1 transmission hologram suitable for the generation of a white-lightreflection hologram a grid a pixels having different vertical andhorizontal packing densities is written as shown in FIG. 7. If areflection type master hologram is required then the system isconfigured to the state shown in FIG. 4. In order to write a directone-step reflection hologram, the basic image data is mathematicallytransformed according to special pixel-swapping rules, the system isconfigured as shown in FIG. 5, and pixels are written as shown in FIG.8.

Turning now to consider the preferred embodiment of the presentinvention, in one embodiment a computer is used to generate a threedimensional model of an object using a standard commercial computerprogram. Current computer programs can produce very lifelike modelsusing a variety of sophisticated rendering processes that mimic reallife effects. In addition advances in computer technology have now seenthe calculation times, required for such programs to run, dramaticallydecreased. Three dimensional scanners using Moiré or other principlesnow permit the incorporation of real world 3-D images in such computermodels. The storage memory required for such 3-D models is largelydependent on the texture maps used therein and hence computer filesrepresenting such 3-D models are usually relatively small and may betransmitted over the internet easily. In the preferred embodiment of theinvention we use such 3-D computer models to generate a series of 2-Dcamera views from a virtual viewing plane as shown in FIG. 9. Here theviewing plane is labeled 901 and individual 2-D perspective cameraimages, such as 905 and 904, of the computer represented object 900 aregenerated at multiple locations on the viewing plane such as 902 and903. The spacing and density of such 2-D views are generally controlledaccording to the information required for a certain type of hologram butin one embodiment they form a regular 2-D matrix and in another aregular horizontal 1-D array.

In another embodiment of the invention a real model is used instead of acomputer representation and a real camera is employed to recordindividual photographs (either digitally or via photographic film thatis subsequently digitized). In such a case FIG. 9 should be interpretedin the following fashion. Object 900 represents the object to beholographed. 901 represents the plane on which a camera 902 ispositioned. Photographs of the object 900 are taken at a variety ofpositions on this plane. For example the view position 906 yields thephotograph 905 and the view position 903 yields the photograph 904.Generally some mechanism is used to transport a camera from position toposition in a sequential fashion often using a 1 or 2 dimensionaltranslation stage to accomplish this. As before, the spacing and densityof such 2-D views are generally controlled according to the informationrequired for a certain type of hologram but in one embodiment they forma regular 2-D matrix and in another a regular horizontal 1-D array.

In both of the above cases restricted animation, which may betransferred to the final hologram, may be modeled by arranging that themodel 900 moves in a defined sense (representing such animation) asdifferent camera positions are selected on the plane 901, such camerapositions following sequential monotonic trajectories on the plane. Onobserving the final hologram, an observer following such sequentialmonotonic trajectory in the observation space will perceive theanimation.

Mathematical Definition of the 3-D Image Data

The preferred embodiment works by defining a set of 2-D views of a realor computer represented object 900 (FIG. 9) on a certain viewing plane901 (for each of several colours) and processing such views digitally togenerate data (e.g. 904, 905) that may then be displayed on spatiallight modulators within the printing device described above.

With reference to FIG. 10, which shows the object 900 being digitallyphotographed, let us define the Cartesian coordinates ξ and ζ torepresent respectively the x and y directions on the camera plane 901.We define the origin of this coordinate system as the bottom left handcorner of 901. Now let us further discretize the plane 901 as follows:

$\begin{matrix}\begin{matrix}{{\xi = {\left( {k - 1} \right)\;\frac{\Xi}{\left( {N_{K} - 1} \right)}}},} & \; & \; & {{k = 1},\ldots\mspace{11mu},N_{K}}\end{matrix} & (1) \\\begin{matrix}{{\zeta = {\left( {g - 1} \right)\;\frac{\Theta}{\left( {N_{G} - 1} \right)}}},} & \; & \; & {{g = 1},\ldots\mspace{11mu},N_{G}}\end{matrix} & (2)\end{matrix}$where the integers k and g label perspective view locations on 901. Agrid of (N_(K)×N_(G)) perspective views is thus envisaged to cover theplane 901 which has dimensions Ξ×Θ.

With reference to FIG. 11 let us now define the Cartesian coordinates xand y to describe each perspective view of size Q_(X)×Q_(Y) (e.g. 904,905). Again we use an origin at the bottom left-hand corner anddiscretize as above:

$\begin{matrix}\begin{matrix}{{x = {\left( {i - 1} \right)\;\frac{Q_{X}}{\left( {N_{I} - 1} \right)}}},} & \; & \; & {{i = 1},\ldots\mspace{11mu},N_{I}}\end{matrix} & (3) \\\begin{matrix}{{y = {\left( {j - 1} \right)\;\frac{Q_{Y}}{\left( {N_{J} - 1} \right)}}},} & \; & \; & {{j = 1},\ldots\mspace{11mu},N_{J}}\end{matrix} & (4)\end{matrix}$

A grid of (N_(I)×N_(J)) pixels is thus envisaged to cover eachperspective view with each such view having dimensions of Q_(X)×Q_(Y).It is useful to project the “perspective-view” plane (904 & 905 etc ofFIG. 11) onto the hologram plane (1201 in FIG. 12).

In the case that we wish to model a full-parallax 3-D scene we definethe luminous intensity tensor ^(kg)I_(ij). This tensor represents thetotality of information describing the 3-D scene. It can either beprovided by multiple photographic data or as the output of a standardcommercial 3-D modeling program. In the case of horizontal parallaxholograms the index g is fixed and we define the luminous intensitytensor as simply ^(k)I_(ij).

Mathematical Definition of the Hologram

FIG. 12 shows the hologram 1201, that we wish to generate, of the 3Dobject 900. The viewing plane of this hologram is now represented by 901and individual viewing locations by, for example, 903 and 906. Clearlywe wish to arrange that when a viewer places his eye at 906 he will seethe image 905. Conversely when the viewer places his eye at 903 heshould see the image 904.

The hologram that will be written is composed of many individualholopixels (FIG. 12), the location of which is described by theCartesian coordinates(X, Y). As before the following notation will beadopted:

$\begin{matrix}\begin{matrix}{{X = {\left( {\alpha - 1} \right)\;\frac{D}{N_{A} - 1}}},} & \; & \; & {{\alpha = 1},\ldots\mspace{11mu},N_{A}}\end{matrix} & (5) \\\begin{matrix}{{Y = {\left( {\beta - 1} \right)\;\frac{R}{N_{B} - 1}}},} & \; & \; & {{\beta = 1},\ldots\mspace{11mu},N_{B}}\end{matrix} & (6)\end{matrix}$where D represents the hologram (horizontal) width and R the hologramheight.

Let us now imagine a viewing plane which is geometrically coincidentwith the camera plane, situated at a distance H from the hologram andlaterally centred. In this case the Cartesian coordinates (ξ,ζ) thathave been defined above may be used for both the treatment of theviewing plane and the camera plane.

In the case of horizontal parallax holograms, the view of the hologramis the same irrespective of the vertical position bar a simple imageshift. In this case a line may be defined, which we will call the cameraline, which is horizontal and which is vertically centred in the viewingplane. If an H1 hologram were being written, this line would coincideexactly with the physical strip-master. Our computer model would thensupply us with N_(K) discrete views at regular intervals along thisline. The variable ζ would no longer be needed and the distance ξ fromthe LHS of the camera line to a given camera view would be given byequation 1.

We now assume that all optical distortion is absent from the printerobjective. This enables us to establish the required formalism for theparaxial case. Later we will then generalize our mathematics to thenon-paraxial finite-distortion case.

FIG. 13 shows a simplified diagram of how we intend to write thehologram, pixel by pixel. Data is written to an SLM (1301). A laser beamilluminates this SLM and is then focused by the objective 1302 to theholopixel 1303 on the hologram 1304 (reference beam not shown). Sincethe objective (1302) is paraxial we can, without loss of generality,project the SLM plane (1301) onto the camera/viewing plane (1305)exactly in the same fashion as we projected the perspective view planeonto the hologram plane above. We thus define the location of a pixel onthe SLM by its (projected) x and y Cartesian coordinates (u, v) on theviewing plane where

$\begin{matrix}\begin{matrix}{{u = {\left( {\mu - 1} \right)\;\frac{\Pi}{N_{M} - 1}}},} & \; & \; & {{\mu = 1},\ldots\mspace{11mu},N_{M}}\end{matrix} & (7) \\\begin{matrix}{{v = {\left( {v - 1} \right)\;\frac{\Sigma}{N_{V} - 1}}},} & \; & \; & {{v = 1},\ldots\mspace{11mu},N_{V}}\end{matrix} & (8)\end{matrix}$

The parameters Π and Σ effectively define respectively the horizontaland vertical fields of view (FOV) of the printer writing head. Theobjective optics are almost always circularly symmetric but whencombined with an SLM unit, we obtain different FOVs in the horizontaland vertical directions. The paraxial printer FOVs are related to theparameters Π and Σ by the following relations:

$\begin{matrix}\begin{matrix}{\Psi_{PH} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{\Pi}{2H} \right\}}} \\{\Psi_{PV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{\Sigma}{2H} \right\}}}\end{matrix} & \left( {8a} \right)\end{matrix}$

Here the subscript P refers to “Printer” and the subscripts “H” and “V”refer respectively to horizontal and vertical. The parameter H, asdiscussed above, is the distance between the hologram and theviewing/camera plane.

To recapitulate, in the case of a full parallax hologram our computerdata will consist of N_(K)×N_(G) perspective views of the requiredhologram image. Generally we will express this information asN_(K)×N_(G) intensity matrices or by using the simple intensity tensor^(kg)I_(ij), which gives the intensity distribution, I, at the cameraview pixel (i, j) for the horizontal camera position k and the verticalcamera position g. In the case of a single (horizontal) parallaxhologram our computer model will consist of ^(k)I_(ij). Ultimately wewish to calculate the data that must be written to the SLM(s) which werefer to as the paraxial mask file intensity distributions and which wewill denote in tensor form as ^(μν)S_(αβ).

Single Parallax Monochrome Reflection Hologram with Static SLM

There are several choices as to how physically a single parallax directwrite hologram may be written. Firstly, we may decide to keep the SLMstatic or we may use an objective having a larger entrance pupil and optto move the SLM within this pupil. Secondly various formats of computerdata that effectively correspond to different (virtual) camerageometries may be used. A simple translating camera will produce adifferent intensity tensor ^(k)I_(ij) than a specially programmed camerafor example. Finally, different viewing window geometries may beselected. If we elect to use the full FOV of the objective when writingeach holopixel then we will have a different result than if we constrainour viewing window to be a well-defined rectangle.

All the above choices must be made according to the specific applicationat hand. In the following sections we will treat the most importantmajor cases and derive for each geometry the pixel-swapping (orinterpolatory) transformations necessary to convert the single parallaxperspective tensor ^(k)I_(ij) into the mask tensor ^(μν)S_(αβ).

Non-Centred Image with Fixed Rectangular Viewing Window of Same Size asHologram

In this section a computer model, ^(k)I_(ij), will be assumed. Thismodel has been derived by a simple translating camera which follows ahorizontal trajectory through the mid-point of the hologram viewingplane (when one interchanges the hologram+viewing plane for the virtualobject+camera plane—see FIG. 14 a). The viewing zone of the hologram isnow defined to be geometrically identical to the actual hologram andthus constitutes a rectangle having horizontal dimension D and verticaldimension R (FIG. 14 b). Hence Ξ=D.

The viewing window is laterally displaced from the hologram by adistance H and hence the camera track is also laterally displaced fromthe virtual object by this same distance H. The (virtual) camera FOV(Ψ_(CH)) is now chosen such that the rectangular view of the camera at adistance H is of width 2D. Therefore, when positioned at the extreme LHSof the viewing line, the camera's view will just extend to the extremeRHS of the hologram and visa-versa as shown in FIG. 14 b. So too thehorizontal FOV of the printer write-head (Ψ_(PH)) is chosen to be thesame as the horizontal camera FOV (Ψ_(CH)). Mathematically,

$\begin{matrix}{{Q = {\Pi = {2D}}},} & {Q_{Y} = R} & {and} & {\Psi_{CH} = {\Psi_{PH} = {2\mspace{11mu}\tan^{- 1}{\left\{ \frac{D}{H} \right\}.}}}}\end{matrix}$Since the hologram is single-parallax the vertical camera FOV is chosenas

$\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{R}{2H} \right\}}$and the printer FOV is chosen as in equation 8a.

Now, at the holopixel defined by (X,Y) (see FIG. 14 b) it can be seenthat the SLM horizontal pixel coordinate is defined by u=D+w for theindicated ray. To find the corresponding intensity information from thevirtual camera perspective views we need to look up that perspectiveview corresponding to ξ=X+w and to select the horizontal (perspectiveview) pixel coordinate defined by x=D−w in this particular view. Clearlywe will select the vertical coordinate by requiring that y=Y. Therefore

$\begin{matrix}{u = {{D + w} = {\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}}}} & (9)\end{matrix}$from whence it is trivial to see that

$\begin{matrix}{w = {{\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} - D}} & (10)\end{matrix}$Also from this equation and the relation ξ=X+w it follows that

$\begin{matrix}{{\left( {k - 1} \right)\;\frac{D}{N_{K} - 1}} = {{\left( {\alpha - 1} \right)\;\frac{D}{N_{A} - 1}} + {\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} - D}} & (11)\end{matrix}$which simplifies to

$\begin{matrix}{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)} - N_{K} + 2}} & (12)\end{matrix}$This equation states that the ray defined by the mask file index μ andemanating from the holopixel whose index coordinates are (α,β)intersects with the camera view defined by the index k as given in thisexpression. To understand which pixel in the perspective view file thisray corresponds to we use the relation X=D−w derived above from whenceit is seen that

$\begin{matrix}\begin{matrix}{{\left( {i - 1} \right)\;\frac{2D}{N_{I} - 1}} = {D - {\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} + D}} \\{= {{\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} + {2D}}}\end{matrix} & (13)\end{matrix}$or more simply

$\begin{matrix}{i = {N_{I} - \frac{\left( {\mu - 1} \right)\left( {N_{I} - 1} \right)}{N_{M} - 1}}} & (14)\end{matrix}$Finally, from the equation y=Y it follows that

$\begin{matrix}{{\left( {j - 1} \right)\;\frac{R}{N_{J} - 1}} = {\left( {\beta - 1} \right)\;\frac{R}{N_{B} - 1}}} & (15)\end{matrix}$or more simply

$\begin{matrix}{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (16)\end{matrix}$

If we wish to limit the vertical FOV of the hologram as a function ofthe holopixel coordinate Y such as to create a viewing window of exactlythe same dimensions as the actual hologram then we will have to imposethe condition that

$\begin{matrix}\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{}_{}^{}{}_{}^{}}} & {{{{when}\mspace{14mu}\frac{\Sigma}{2}} - Y} \leq v \leq {\frac{\Sigma}{2} - Y + R}} \\{= 0} & {{otherwise}\mspace{14mu}\left( {0 \equiv {{zero}\mspace{14mu}{brightness}}} \right)}\end{matrix} & (17)\end{matrix}$where, of course, Σ≧2R. This condition on v translates into thefollowing condition on ν:

$\begin{matrix}{{{\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}}} \right\}\left( {N_{V} - 1} \right)} + 1} \leq v \leq {{\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}} + \frac{R}{\Sigma}} \right\}\left( {N_{V} - 1} \right)} + 1}} & (18)\end{matrix}$And thus the full pixel-swapping transformation may be written asfollows:

$\begin{matrix}\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{}_{}^{}{}_{}^{}}} & {{{{when}\mspace{14mu}\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}}} \right\}\left( {N_{V} - 1} \right)} + 1} \leq} \\ & {{~~~~~~~~~~~~~~~}{v \leq {{\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}} + \frac{R}{\Sigma}} \right\}\left( {N_{V} - 1} \right)} + 1}}} \\ & {{{and}\mspace{14mu} 0} < k \leq N_{K}} \\{= 0} & {otherwise}\end{matrix} & (19) \\{where} & \; \\{{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)} - N_{K} + 2}},} & (20) \\{{i = {N_{I} - \frac{\left( {\mu - 1} \right)\left( {N_{I} - 1} \right)}{N_{M} - 1}}},} & (21) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (22)\end{matrix}$

In the case that no restriction is imposed on ν then we have the case ofa rolling vertical window with every holopixel having an identicalvertical FOV given by

$\begin{matrix}{\Psi_{HV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{\Sigma}{2H} \right\}}} & (23)\end{matrix}$

Note that this is the same as the paraxial (vertical) printer FOV ofequation 8a

$\begin{matrix}{\Psi_{PV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{\Sigma}{2H} \right\}}} & \left( {23a} \right)\end{matrix}$but it is generally different from the virtual (vertical) camera FOV

$\begin{matrix}{\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{R}{2H} \right\}}} & \left( {23b} \right)\end{matrix}$

In the case of no restriction on ν we may write the pixel swaptransformation more simply as:

$\begin{matrix}\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{}_{}^{}{}_{}^{}}} & {\forall{{v\mspace{14mu}{when}\mspace{14mu} 0} < k \leq N_{K}}} \\{= 0} & {otherwise}\end{matrix} & (24) \\{where} & \; \\{{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)} - N_{K} + 2}},} & (25) \\{{i = {N_{I} - \frac{\left( {\mu - 1} \right)\left( {N_{I} - 1} \right)}{N_{M} - 1}}},} & (26) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (27)\end{matrix}$These equations define how the digital computer model ^(k)I_(ij)consisting of conventional perspective views is converted into the maskfile information ^(μν)S_(αβ) that is necessary to write the holopixelsone by one via the SLM.

We have implicitly assumed that all the parameters i,j,k,g,α,β,μ and νare integers. Generally this does not necessarily have to be the caseand the above transforms are perfectly valid if we insist on a rationalrepresentation. Indeed the transformations given above generallyrequire, in the calculation of ^(μν)S_(αβ) with integer parameters α,β,μand ν, a knowledge of ^(k)I_(ij) with rational values of k, i and j.Such a situation means that either a computer perspective model must begenerated on a specialized mesh that leads to a uniform mesh once thedata is transformed to mask space or multi-dimensional interpolation isused to calculate the values of ^(k)I_(ij); with the required rationalvalues of k, i and j. Both solutions are practical and the choice ofwhich method to use depends on the perspective model software and thecomputer hardware. There is, however, a third solution to the aboveproblem where all index parameters are integers. For instance, if wechoose N_(M) to be odd and we further require

$\begin{matrix}\begin{matrix}{N_{K} = {N_{A} = \frac{N_{M} + 1}{2}}} \\{N_{I} = N_{M}} \\{N_{J} = N_{B}}\end{matrix} & \left( {28\text{-}30} \right)\end{matrix}$then it follows that the transformations given above remain closed inthe integer set. In such a case simplified mask transformations may bewritten. For the case of a well defined rectangular viewing window wemay therefore write:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{{}_{}^{}{}_{}^{}}\mspace{14mu}{when}\mspace{14mu}\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}}} \right\}\left( {N_{V} - 1} \right)} + 1} \leq}} & (31) \\{\mspace{205mu}{v \leq {{\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}} + \frac{R}{\Sigma}} \right\}\left( {N_{V} - 1} \right)} + 1}}} & \; \\{\mspace{130mu}{{{and}\mspace{14mu} 0} \leq k \leq N_{K}}} & \; \\{\mspace{59mu}{= {0\mspace{31mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {\mu + \alpha - N_{K}}},} & (32) \\{{i = {N_{M} - \mu + 1}},} & (33) \\{j = \beta} & (34)\end{matrix}$and in the case of no restriction on ν:^(μν) S _(αβ)=^(k) I _(ij)0≦k≦N _(K)=0 otherwise  (35)where k=μ+α−N _(K),  (36)i=N _(M)−μ+1,  (37)j=β  (38)Centred Image with Fixed Rectangular Viewing Window

In the above section the case of a computer perspective model which wasgenerated by a simply translating camera has been treated. However, sucha model is rather inefficient as 2× as much rendering data must becalculated as is actually required for the case of a hologram viewingwindow equal to the hologram size. For more realistic cases, as shall beshown further on, the factor is even greater than 2×. It is thus rathermore logical to render only that part of the data that actually appearson the hologram. This may be accomplished (see FIG. 15) by usingeffectively the same virtual camera geometry as above but by introducingcamera stops outside of which no rendering is done. This means that weonly render a sliding window in the full field of our virtual camera.Thus we set

$\begin{matrix}\begin{matrix}{\Psi_{CH} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{D}{H} \right\}}} \\{\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{R}{2H} \right\}}}\end{matrix} & \left( {39\text{-}40} \right)\end{matrix}$where Ψ_(CH) is the horizontal camera field-of-view and Ψ_(CV) is thevertical camera field-of-view and we render only between ξ=0 and ξ=D. Inthis way the coordinate point x=0 of every perspective view coincideswith x=0 and we may use Q_(X)=D instead of Q_(X)=2D as in the previoussection. Thus, in FIG. 12 points 1102-1105 on the perspective view plane905 correspond to points 1102-1105 on the hologram plane 1101.

Proceeding as before and with reference this time to FIG. 16 (note thathere Ψ_(PH)=Ψ_(CH)), at the holopixel defined by (X,Y) it is seen thatu=D+w. To find the corresponding intensity information from the virtualcamera perspective views it is necessary to look up, also as before, theperspective view corresponding to ξ=X+w. However the horizontal pixelcoordinate defined by x=X must now be selected in this particular view.The vertical coordinate is selected as always by requiring that y=Y.

Thus

$\begin{matrix}{u = {{D + w} = {\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}}}} & (41)\end{matrix}$from whence trivially it follows that

$\begin{matrix}{w = {{\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} - D}} & (42)\end{matrix}$Also from this equation and the relation ξ=X+W it follows that

$\begin{matrix}{{\left( {k - 1} \right)\frac{D}{N_{K} - 1}} = {{\left( {\alpha - 1} \right)\frac{D}{N_{A} - 1}} + {\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} - D}} & (43)\end{matrix}$which simplifies to

$\begin{matrix}{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{N_{A} - 1} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{N_{M} - 1} - N_{K} + 2}} & (44)\end{matrix}$

This equation states that the ray defined by the mask file index μ andemanating from the holopixel whose index coordinates are (α,β)intersects with the camera view defined by the index k as given in thisexpression. To understand which pixel in the perspective view file thisray corresponds to, the relation x=X, derived above, is used—whence

$\begin{matrix}{{\left( {i - 1} \right)\;\frac{D}{N_{I} - 1}} = {\left( {\alpha - 1} \right)\;\frac{D}{N_{A} - 1}}} & (45)\end{matrix}$or more simply

$\begin{matrix}{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}} & (46)\end{matrix}$Finally, as before, from the equation y=Y it follows that

$\begin{matrix}{{\left( {j - 1} \right)\;\frac{R}{N_{J} - 1}} = {\left( {\beta - 1} \right)\frac{R}{N_{B} - 1}}} & (47)\end{matrix}$or more simply

$\begin{matrix}{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (48)\end{matrix}$Thus, for a proper rectangular viewing window the followingtransformation is derived:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{{}_{}^{}{}_{}^{}}\mspace{14mu}{when}\mspace{14mu}\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}}} \right\}\left( {N_{V} - 1} \right)} + 1} \leq}} & (49) \\{\mspace{205mu}{v \leq {{\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}} + \frac{R}{\Sigma}} \right\}\left( {N_{V} - 1} \right)} + 1}}} & \; \\{\mspace{130mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{34mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)} - N_{K} + 2}},} & (50) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (51) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (52)\end{matrix}$

In the case that no restriction is imposed on ν then we have the case ofa rolling vertical window with every holopixel having an identicalvertical FOV. In this case the pixel swap transformation may be writtenas:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{}_{}^{}{}_{}^{}}\mspace{20mu}{\forall{{v\mspace{14mu}{when}\mspace{14mu} 0} < k \leq N_{K}}}}} & (53) \\{\mspace{56mu}{= {0\mspace{45mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)} - N_{K} + 2}},} & (54) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (55) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (56)\end{matrix}$

As before certain constraints can be identified that restrict the abovetransformations to the integer set. Thus, for example, we may requireN_(M) to be odd as before and we may further require

$\begin{matrix}\begin{matrix}{N_{K} = {\frac{N_{M} + 1}{2} = {N_{A} = N_{I}}}} \\{N_{J} = N_{B}}\end{matrix} & \left( {57\text{-}58} \right)\end{matrix}$In such a case a simplified mask transformation may be written where allindices are integer. For the case of a well defined rectangular viewingwindow it follows that:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{{}_{}^{}{}_{}^{}}\mspace{14mu}{when}\mspace{14mu}\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}}} \right\}\left( {N_{V} - 1} \right)} + 1} \leq}} & (59) \\{\mspace{205mu}{v \leq {{\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma}} + \frac{R}{\Sigma}} \right\}\left( {N_{V} - 1} \right)} + 1}}} & \; \\{\mspace{130mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{34mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {\mu + \alpha - N_{K}}},} & (60) \\{{i = \alpha},} & (61) \\{j = \beta} & (62)\end{matrix}$and for the case of no restriction on ν^(μν) S _(αβ)=^(k) I _(ij) ∀ν when 0<k≦N _(K)=0 otherwise  (63)wherek=μ+α−N _(K) , i=α, j=β.  (64-66)Centred Image with Maximum FOV

In the previous two sections the horizontal size of the viewing windowhas been restricted to be the same physical dimension as the actualhologram. The advantage of doing this is that in the viewing zone youeither see all the hologram or nothing. However, each holopixel ispotentially capable of replaying a fixed FOV. If all of this FOV were tobe used, the effect would be a scrolling horizontal window instead of anabrupt image termination (assuming of course that the observer islocated at the viewing plane). This case may be treated by considering acamera track longer than previously considered. Hence Ξ==D+T. We furtherset Π=T (defining the horizontal printer FOV). Finally, we set Q_(X)=Dand Q_(Y)=R (defining the centred camera). As in the section entitled“Centred Image with Fixed Rectangular Viewing Window” a centred camera(FIG. 17) is considered and we choose

$\begin{matrix}\begin{matrix}{\Psi_{CH} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{T}{2H} \right\}}} \\{\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{R}{2H} \right\}}}\end{matrix} & \left( {67\text{-}68} \right)\end{matrix}$where Ψ_(CH) is the horizontal camera field-of-view, Ψ_(CV) is thevertical camera field-of-view and rendering is performed only betweenX=0 and X=D.

As before and with reference this time to FIG. 18, at the holopixeldefined by(X,Y) it is seen that u=T/2+W. To find the correspondingintensity information from the virtual camera perspective views it isnecessary now to look up the perspective view corresponding toξ=T/2+X+w. And as in the section entitled “Centred Image with FixedRectangular Viewing Window” it is necessary to select the horizontalpixel coordinate defined by x=X in this particular view. The verticalcoordinate is selected as always by requiring that y=Y. Thus ourexpression (equation 10) for w becomes

$\begin{matrix}{w = {{u - \frac{T}{2}} = {{\left( {\mu - 1} \right)\;\frac{T}{N_{M} - 1}} - \frac{T}{2}}}} & (69)\end{matrix}$Likewise the new equation ξ=T2+X+w now leads to:

$\begin{matrix}{{\left( {k - 1} \right)\;\frac{D + T}{N_{K} - 1}} = {\frac{T}{2} + {\left( {\alpha - 1} \right)\frac{D}{N_{A} - 1}} + {\left( {\mu - 1} \right)\;\frac{T}{N_{M} - 1}} - \frac{T}{2}}} & (70)\end{matrix}$which simplifies to

$\begin{matrix}{k = {{\frac{D}{D + T}\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{N_{A} - 1}} + {\frac{T}{D + T}\frac{\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{N_{M} - 1}} + 1}} & (71)\end{matrix}$

The equations for i and j remain as in the section entitled “CentredImage with Fixed Rectangular Viewing Window” and hence the final masktransformation for maximum FOV may be written:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{}_{}^{}{}_{}^{}}\mspace{20mu}{\forall{{v\mspace{14mu}{when}\mspace{14mu} 0} < k \leq N_{K}}}}} & (72) \\{\mspace{59mu}{= {0\mspace{40mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {{\frac{D}{D + T}\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{N_{A} - 1}} + {\frac{T}{D + T}\frac{\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{N_{M} - 1}} + 1}},} & (73) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (74) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (75)\end{matrix}$

If we wanted to limit the vertical viewing window to be of a fixedheight Γ whilst keeping a maximum horizontal FOV, we could also use (seeFIG. 19):

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{14mu}{when}\mspace{14mu} v} \geq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R - \Gamma}{2\Sigma}} \right\}} + 1}}} & (76) \\{\mspace{130mu}{{{and}\mspace{14mu} v} \leq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R + \Gamma}{2\Sigma}} \right\}} + 1}}} & \; \\{\mspace{130mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{34mu}{otherwise}}}} & (77) \\{where} & \; \\{{k = {{\frac{D}{D + T}\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{N_{A} - 1}} + {\frac{T}{D + T}\frac{\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{N_{M} - 1}} + 1}},} & (78) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (79) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (80) \\{with} & \; \\{\Sigma \geq {R + \Gamma}} & (81)\end{matrix}$

In passing it should be noted that this condition may equally well beapplied to the transformations of equations 19,31,49 and 59 if ourintention is to restrict the vertical window to a height Γ. One may wantto do this, by example, to achieve a greater hologram brightness.

Rectangular Viewing Window of General Size

In the above sections the case of a rectangular window of the same sizeas the hologram and the case of a scrolling window of unrestrained fieldof view have been treated. In this section, for completeness, the caseof an arbitrarily sized viewing window and an arbitrary recording FOVwill be treated.

In our above discussions of rectangular windows certain key parameters,such as the hologram recording FOV and the distance from the viewingplane to the hologram, were chosen in such a manor as to use mostefficiently the printer SLM. However, under real conditions it ispossible that to restrain Π to be equal to exactly twice the size of thehologram, for instance, as we have discussed in our first two analyses,may at some times be inconvenient. If a rectangular viewing window ofhorizontal dimension L and vertical dimension Γ is considered then ourprevious discussions can be generalized to requiring only that Π≧D+L andthat Σ≧R+Γ. A centred camera will be used as in the two previoussections above and so we will require that Q_(X)=D and Q_(Y)=R. Sincethe horizontal dimension of the viewing window is L we shall requireΞ=L.

As before it should be noted that the recording or printer FOVs aredefined by Π and Σ. The camera FOVs however are now somewhat different.We will require that

$\begin{matrix}\begin{matrix}{\Psi_{CH} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{D + L}{2H} \right\}}} \\{\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{R}{2H} \right\}}}\end{matrix} & \left( {81a} \right)\end{matrix}$where, as before, Ψ_(CH) is the horizontal camera field-of-view, Ψ_(CV)is the vertical camera field-of-view and rendering is performed onlybetween X=0 and X=D.

With reference to FIG. 20, it is seen that u=Π/2+w. The correspondingequation for ξ is now ξ=(L−D)/2+X+w and the equation for x is x=X. Asusual the equation y=Y remains unchanged. These equations now tell usthe rules for k, i and j:

$\begin{matrix}{k = {{\left\{ \frac{N_{K} - 1}{L} \right\}\left\{ {\frac{L - D}{2} + \frac{\left( {\alpha - 1} \right)D}{N_{A} - 1} + \frac{\left( {\mu - 1} \right)\Pi}{N_{M} - 1} - \frac{\Pi}{2}} \right\}} + 1}} & (82) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (83) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (84)\end{matrix}$The vertical window is dwelt with exactly as previously and hence thegeneral mask transformation may be written as:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{14mu}{when}\mspace{14mu} v} \geq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R - \Gamma}{2\Sigma}} \right\}} + 1}}} & (85) \\{\mspace{130mu}{{{and}\mspace{14mu} v} \leq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R + \Gamma}{2\Sigma}} \right\}} + 1}}} & \; \\{\mspace{130mu}{{{and}\mspace{14mu} 0} \leq k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{34mu}{otherwise}}}} & (86) \\{where} & \; \\{{k = {{\left\{ \frac{N_{K} - 1}{L} \right\}\left\{ {\frac{L - D}{2} + \frac{\left( {\alpha - 1} \right)D}{N_{A} - 1} + \frac{\left( {\mu - 1} \right)\Pi}{N_{M} - 1} - \frac{\Pi}{2}} \right\}} + 1}},} & (87) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (88) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (89)\end{matrix}$

In passing we should note that usually one should try and arrange thatΠ=D+L in order to utilize the full horizontal resolution of the SLM.

Moving SLM, Centred Camera and General Rectangular Window

Up until now the case of a static SLM which remains fixed in theentrance pupil of the writing objective has been considered. However,when we wish to make a hologram with a well defined rectangular window,we may do so by moving the SLM within the entrance pupil of theobjective. In this case, of course, the objective must have a largerentrance pupil relative to the SLM size in order to accommodate suchmotion. One of the principle advantages of this scheme is its use indual-mode printers. Generally MW printers that write H1 type masterholograms require a moving SLM (in order to use effectively the SLMresolution) and hence, by using this scheme in a dual function printer,exactly the same print-head assembly can be employed to write either a1-step hologram or an H1 hologram master. This is not the case if we usea static SLM.

Let us assume as usual that our hologram is of horizontal size D andvertical size R and that our viewing window, displaced laterally fromthe hologram by a distance H, is of horizontal dimension L and verticaldimension Γ. If the maximum angular resolution possible is to beattained with our SLM, H must be chosen with reference to the FOV of thewriting objective such that the virtual image of the SLM at the viewingplane is of horizontal dimension L.

Now one can choose to move the SLM only horizontally, or one can chooseto move it in a two-dimensional fashion. If it is moved 2-dimensionallythen all the vertical pixels of the SLM will be used only in the casethat the vertical SLM image size in the viewing plane is Γ. If thevertical size is restricted such that the aspect ratio of the SLM nolonger corresponds to the aspect ratio of the viewing window thengenerally we will only use a fixed percentage of the vertical SLM heightand Σ≧Γ. In the case that we elect to only move the SLM horizontally andto permanently position it in the vertical centre of the objective pupilthen either an unrestricted scrolling window can be used or ν can berestrained, as in the previous sections, so as to create a fixedrectangular viewing window.

Firstly it should be pointed out that for the case of a horizontalviewing window size of L, the writing objective must have an FOV of atleast

$\begin{matrix}{\Psi_{\min} = {2\mspace{11mu}{\tan^{- 1}\left( \frac{L + D}{2H} \right)}}} & (90)\end{matrix}$In the unlikely event that the viewing window is taller than it is widerthen we would require that

$\begin{matrix}{\Psi_{\min} = {2\mspace{11mu}{\tan^{- 1}\left( \frac{\Gamma + R}{2H} \right)}}} & (91)\end{matrix}$but it should be noted that this is rather unlikely in the context of asingle parallax hologram. It should also be noted that these equationsconstitute absolute limits pertaining to the translation of the SLMwithin the objective pupil in either a strictly horizontal or a strictlyvertical fashion. Slightly more severe criteria may be derived byexamining the outer diagonal points of the SLM within the objectivepupil.

Let us consider a 2-D displacement of the SLM by denoting the x and ycoordinates of the centre of the projected image of the SLIM at theviewing plane by τ and η. We choose the origin of this coordinate systemat the holopixel being written—namely(X,Y). FIGS. 21 and 22 show thegeometry from respectively a top and side view. In order to keep theimage of the SLM at a fixed location on the viewing plane as indicatedin the diagrams we will move it according to the law

$\begin{matrix}\begin{matrix}{\tau = {{\frac{D}{2} - X} = {\frac{D}{2} - {\left( {\alpha - 1} \right)\frac{D}{N_{A} - 1}}}}} \\{\eta = {{\frac{R}{2} - Y} = {\frac{R}{2} - {\left( {\beta - 1} \right)\frac{R}{N_{B} - 1}}}}}\end{matrix} & \left( {92\text{-}93} \right)\end{matrix}$Note that the actual x and y coordinates ({hacek over (τ)} and {hacekover (η)}) of the centre of the SLM relative to the centre of theparaxial objective are just linearly proportional to τ and η (i.e.{hacek over (τ)}=aτ and {hacek over (η)}=aη where the constant a isrelated to the characteristics (magnification) of the objective—see FIG.22 a). Since a centred virtual camera is being considered we requireQ_(X)=D and Q_(Y)=R. Since our viewing window is of size (L×Γ) we willalso require that Ξ=Π=L and that Σ≧Γ.

From FIGS. 21 and 22 we see immediately that u=ξ, x=X and y=Y. Also fromFIG. 22 we see that in order to limit the vertical window size to Γ≦Σ wemust ensure that

$\begin{matrix}{\frac{\Sigma - \Gamma}{2} \leq v \leq \frac{\Sigma + \Gamma}{2}} & (94)\end{matrix}$Thus we may write the mask transformation for the case of a 2-D movingSLM, centred virtual camera and of general rectangular viewing area as

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu}\frac{\Sigma - \Gamma}{2\Sigma}\left( {N_{V} - 1} \right)} + 1} \leq}} & (95) \\{\mspace{211mu}{v \leq {{\frac{\Sigma + \Gamma}{2\Sigma}\left( {N_{V} - 1} \right)} + 1}}} & \; \\{\mspace{135mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{40mu}{otherwise}}}} & (96) \\{where} & \; \\{{k = {\frac{\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{N_{M} - 1} + 1}},} & (97) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (98) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (99)\end{matrix}$

If we do not wish to limit the hologram vertical FOV and hence to have avertical scrolling window, then we may decide to fix the SLM verticallyin the objective mid-plane and move it only horizontally. The abovetransform will then also apply with the exception'that there is norestriction on Γ.

If we demand that N_(K)=N_(M), N_(I)=N_(A) and N_(J)=N_(B) we see thatthe rules for k, i and j become trivially k=μ,m i=α and j=β.

Full Parallax Monochrome Reflection Hologram with Static SLM

Transformations derived for the single parallax case may be generalizedwith the greatest of ease to the full parallax case.

Non-Centred Image with Fixed Rectangular Viewing Window of Same size asHologram for the Full-Parallax Case

We should first point out that a non-centred virtual camera of fixed FOVand which only translates in two dimensions provides a poor solution toany real design. This is because the computer is forced to generate atleast four times the perspective information than is actually required.

As we have discussed before, FIG. 14 a shows a view, from the top, ofthe camera tracking from left to right. Note that the camera alwayspoints straight ahead and that the camera FOV remains fixed. FIG. 14 bshows the relevant geometry for the horizontal parallax. Since generallythe aspect ratio of the SLM will not be the same as the aspect ratio ofthe hologram viewing zone, the side view looks a little different and isshown in FIG. 23.

Since the horizontal viewing width is D we must require that Ξ=D. Sincethe vertical viewing height is R we likewise require that Θ=R. Ourchoice of a translating virtual camera imposes the constraints Q_(X)=2Dand Q_(Y)=2R and finally we will choose H such that Π=2D. Note that Σ≧2Ras generally the aspect ratios of SLM and viewing zone will not be thesame.

FIG. 14 b and the logic of the section entitled “Non-Centred Image withFixed Rectangular Viewing Window of Same Size as Hologram” can now beused to derive transformation rules for the indices k and i. These rulesare identical to those given in equations 12 and 14. Referring now toFIG. 23 we see the relations u=D+w, ξ=X+w and x=D−w in the horizontalplane are now replaced with

$\begin{matrix}{{v = {\frac{\Sigma}{2} + w}},} & {ϛ = {Y + w}} & {and} & {y = {\frac{\Sigma}{2} - w}}\end{matrix}$in the vertical plane. These relations give us equations for g and j:

$\begin{matrix}\begin{matrix}{g = {{\frac{N_{G} - 1}{R}\left\{ {\frac{R\left( {\beta - 1} \right)}{N_{B} - 1} + \frac{\Sigma\left( {v - 1} \right)}{N_{V} - 1} - \frac{\Sigma}{2}} \right\}} + 1}} \\{j = {{\frac{\Sigma}{2R}\left\{ \frac{\left( {N_{J} - 1} \right)\left( {N_{V} - v} \right)}{\left( {N_{V} - 1} \right)} \right\}} + 1}}\end{matrix} & \left( {100\text{-}101} \right)\end{matrix}$Thus the full-parallax translating camera mask transformation may bewritten as

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} 0} < g \leq N_{G}}} & (102) \\{\mspace{135mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{40mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)} - N_{K} + 2}},} & (103) \\{{i = {N_{I} - \frac{\left( {\mu - 1} \right)\left( {N_{I} - 1} \right)}{N_{M} - 1}}},} & (104) \\{{g = {{\frac{N_{G} - 1}{R}\left\{ {\frac{R\left( {\beta - 1} \right)}{N_{B} - 1} + \frac{\Sigma\left( {v - 1} \right)}{N_{V} - 1} - \frac{\Sigma}{2}} \right\}} + 1}},} & (105) \\{j = {{\frac{\Sigma}{2R}\left\{ \frac{\left( {N_{J} - 1} \right)\left( {N_{V} - v} \right)}{\left( {N_{V} - 1} \right)} \right\}} + 1.}} & (106)\end{matrix}$

In the case that Σ=2R and both N_(M) and N_(V) are odd and further

$\begin{matrix}\begin{matrix}{N_{K} = {N_{A} = \frac{N_{M} + 1}{2}}} \\{N_{I} = N_{M}} \\{N_{G} = {N_{B} = \frac{N_{V} + 1}{2}}} \\{N_{J} = N_{V}}\end{matrix} & \left( {107\text{-}110} \right)\end{matrix}$a much simplified transform may be written:^(μν) S _(αβ)=^(kg) I _(ij) when 0<g≦N _(G) and 0<k≦N _(K)=0otherwise  (111)wherek=μ+α−N _(K),  (112)i=N _(M)−μ+1,  (113)g=ν+β−N _(G),  (114)j=N _(V)−ν+1.  (115)Centred Image with Fixed Rectangular Viewing Window of same size asHologram for the Full Parallax Case

We now generalize the concept of the centred camera to two dimensions.We thus change the camera apertures such that only data obeying theconstraints 0≦X≦D and 0≦Y≦R is rendered. As usual we will choose H suchthat Π=2D.

The equations governing k and i remain as in the section entitled“Centred Image with Fixed Rectangular Viewing Window”.

However, as in the previous section, since the aspect ratio'of thehologram viewing zone will not in general be the same as the aspectratio of the SLM, the vertical view needs more care (see FIG. 24).

We now set Θ=R=Q_(y). We see that the single-parallax centred-camerarelations u=D+w, ξ=X+w and x=X in the horizontal plane are now replacedwith

${v = {\frac{\Sigma}{2} + w}},$ζ=Y+w and y=Y in the vertical plane. These relations give us the newequations for g and j:

$\begin{matrix}{g = {{\frac{N_{G} - 1}{R}\left\{ {\frac{R\left( {\beta - 1} \right)}{N_{B} - 1} + \frac{\Sigma\left( {v - 1} \right)}{N_{V} - 1} - \frac{\Sigma}{2}} \right\}} + 1}} & (116) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & (117)\end{matrix}$The full-parallax centred-camera mask transformation may now be writtenas

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} 0} < g \leq N_{G}}} & (118) \\{\mspace{146mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{50mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + \frac{2\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)} - N_{K} + 2}},} & (119) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (120) \\{g = {{\frac{N_{G} - 1}{R}\left\{ {\frac{R\left( {\beta - 1} \right)}{N_{B} - 1} + \frac{\Sigma\left( {v - 1} \right)}{N_{V} - 1} - \frac{\Sigma}{2}} \right\}} + 1}} & (121) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & (122)\end{matrix}$

In the case that Σ=2R and both N_(M) and N_(V) are odd and further

$\begin{matrix}{N_{K} = {N_{A} = {N_{I} = \frac{N_{M} + 1}{2}}}} & (123) \\{N_{G} = {N_{B} = {N_{J} = \frac{N_{V} + 1}{2}}}} & (124)\end{matrix}$a much simplified transform may be written:^(μν) S _(αβ)=^(kg) I _(ij) when 0<g≦N _(G) and 0<k≦N _(K)=0otherwise  (125)wherek=μ+α−N _(K)  (126)i=μ  (127)g=ν+β−N _(G)  (128)j=ν  (129)The Maximum FOV Case for Full-Parallax Holograms

We now generalize the section entitled “Centred Image with Maximum FOV”to the case of horizontal and vertical scrolling windows of maximum FOV.With reference to FIG. 17 we choose the camera FOV to be

$\begin{matrix}\begin{matrix}{\Psi_{CH} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{T}{2H} \right\}}} \\{\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{\Sigma}{2H} \right\}}}\end{matrix} & \left( {130\text{-}131} \right)\end{matrix}$and we choose to render only those points obeying the constraints 0≦X≦Dand 0≦Y≦R. We choose H such that π=T (i.e. one chooses H so as to arriveat the required width of the viewing zone=D+T which of course depends onthe horizontal printer

$\left. {{FOV} = {\Psi_{PH} = {2\mspace{11mu}\tan^{- 1}\left\{ \frac{\Pi}{2H} \right\}}}} \right).$Other parameters are Ξ=D+T, Θ=R+Σ, Q_(X)=D and Q_(Y)=R. The equationsfor the horizontal plane are therefore as before. FIG. 25 depicts thevertical situation from which it can be seen that the key equations are

${v = {\frac{\Sigma}{2} + w}},{ϛ = {{\frac{\Sigma}{2} + Y + {w\mspace{14mu}{and}\mspace{14mu} y}} = {Y.}}}$The mask transformation for maximum FOV may therefore be written as:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} 0} < g \leq N_{G}}} & (132) \\{\mspace{146mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{56mu}{= {0\mspace{50mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {{\left\lbrack \frac{D}{D + T} \right\rbrack\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)}} + {\left\lbrack \frac{T}{D + T} \right\rbrack\frac{\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)}} + 1}},} & (133) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (134) \\{g = {{\left\lbrack \frac{R}{R + \Sigma} \right\rbrack\frac{\left( {\beta - 1} \right)\left( {N_{G} - 1} \right)}{N_{B} - 1}} + {\left\lbrack \frac{\Sigma}{R + \Sigma} \right\rbrack\frac{\left( {v - 1} \right)\left( {N_{G} - 1} \right)}{N_{V} - 1}} + 1}} & (135) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & (136)\end{matrix}$

As before, if we wanted to write a hologram with the maximum horizontalFOV but to limit the vertical window to a fixed height, Γ, then we couldapply the conditional transform of equation 76. Or, in other words:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} v} \geq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R - \Gamma}{2\Sigma}} \right\}} + 1}}} & (137) \\{\mspace{146mu}{{{and}\mspace{14mu} v} \leq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R + \Gamma}{2\Sigma}} \right\}} + 1}}} & \; \\{\mspace{146mu}{{{and}\mspace{14mu} 0} < k \leq N_{K}}} & \; \\{\mspace{146mu}{{{and}\mspace{14mu} 0} < g \leq N_{G}}} & \; \\{\mspace{59mu}{= {0\mspace{45mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {{\left\lbrack \frac{D}{D + T} \right\rbrack\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)}} + {\left\lbrack \frac{T}{D + T} \right\rbrack\frac{\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)}} + 1}},} & (138) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (139) \\{g = {{\left\lbrack \frac{R}{R + \Sigma} \right\rbrack\frac{\left( {\beta - 1} \right)\left( {N_{G} - 1} \right)}{N_{B} - 1}} + {\left\lbrack \frac{\Sigma}{R + \Sigma} \right\rbrack\frac{\left( {v - 1} \right)\left( {N_{G} - 1} \right)}{N_{V} - 1}} + 1}} & (140) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & (141)\end{matrix}$This would use the same image data as was used in equations 132 to 136but would simply convert the vertical scrolling window into arectangular window.

An easier way to limit the vertical window to a fixed height, Γ would beto re-render and incorporate the image clipping within the g index swaprule. In this case Θ=Γ which would be considerably more(computationally) efficient over the previous criteria that Θ=R+Σ. Inthis case the mask transform becomes

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} 0} < k \leq N_{K}}} & \left( {141a} \right) \\{\mspace{146mu}{{{and}\mspace{14mu} 0} < g \leq N_{G}}} & \; \\{\mspace{59mu}{= {0\mspace{45mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {{\left\lbrack \frac{D}{D + T} \right\rbrack\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)}} + {\left\lbrack \frac{T}{D + T} \right\rbrack\frac{\left( {\mu - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)}} + 1}},} & \left( {141b} \right) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & \left( {141c} \right) \\{g = {{\left\{ \frac{N_{G} - 1}{\Gamma} \right\}\left\{ {\frac{\Gamma - R}{2} + \frac{\left( {\beta - 1} \right)R}{N_{B} - 1} + \frac{\left( {v - 1} \right)\Sigma}{N_{V} - 1} - \frac{\Sigma}{2}} \right\}} + 1}} & \left( {141d} \right) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & \left( {141e} \right)\end{matrix}$General Rectangular Viewing Zone for Full Parallax

Using exactly the same logic, it is trivially clear that the singleparallax case of a general rectangular window may be generalized to fullparallax. Assuming again a centred camera with Q_(X)=D and Q_(Y)=R and arendering window defined by Ξ=L and Θ=Γ, the mask transformation maythen be written:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} 0} < k \leq N_{K}}} & \left( {142a} \right) \\{\mspace{146mu}{{{and}\mspace{14mu} 0} < g \leq N_{G}}} & \; \\{\mspace{56mu}{= {0\mspace{50mu}{otherwise}}}} & \; \\{where} & \; \\{k = {{\left\{ \frac{N_{K} - 1}{L} \right\}\left\{ {\frac{L - D}{2} + \frac{\left( {\alpha - 1} \right)D}{N_{A} - 1} + \frac{\left( {\mu - 1} \right)\Pi}{N_{M} - 1} - \frac{\Pi}{2}} \right\}} + 1}} & \left( {142b} \right) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & \left( {142c} \right) \\{g = {{\left\{ \frac{N_{G} - 1}{\Gamma} \right\}\left\{ {\frac{\Gamma - R}{2} + \frac{\left( {\beta - 1} \right)R}{N_{B} - 1} + \frac{\left( {v - 1} \right)\Sigma}{N_{V} - 1} - \frac{\Sigma}{2}} \right\}} + 1}} & \left( {142d} \right) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & \left( {142e} \right)\end{matrix}$

The Case of a Moving SLM for Full-Parallax Holograms In the light of theabove sections it will be clear to a person skilled in the art how thesingle parallax analysis already presented for a moving SLM may begeneralized to the full-parallax case.

Offset Window Geometry

In all of the above discussions the viewing window, whether of a fixedrectangular topology or whether of a scrolling nature has beencontemplated to be centrally placed in front of the hologram. However,in a commercial printer that must print large holograms, it is likelythat sub-sections of the final hologram will have to be printedindividually and then assembled. Hence we might consider dividing a 1m×1 m hologram up into 4 pieces of 50 cm×50 cm each. Clearly adiscussion of the printing of these sub-segments entails a discussion ofa general viewing window being offset from the respective hologramsegment.

Due to the importance of this topic two explicit examples will bepresented—one for the single parallax case and one for the doubleparallax case. It will then be evident to someone skilled in the art howthis technique may be generalized to the various other viewing andrecording geometries hereto above presented.

The first example that will be discussed in detail relates to a singleparallax sub-hologram of size D×R with a generalized rectangular viewingzone of size L×Γwhose centre is generally laterally offset from thecentre of the hologram by ω_(x) in the x direction and by ω_(y) in the ydirection. FIGS. 25 a and 25 b show respectively the horizontal andvertical geometries. The visual data for the hologram are generated by acentred camera with the following FOVs (see FIGS. 25 a and 25 b):

$\begin{matrix}\begin{matrix}{\Psi_{CH} = {2\mspace{11mu}\tan^{- 1}\left\{ {\frac{D + L}{2H} + \frac{\omega_{x}}{H}} \right\}}} \\{\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ {\frac{R}{2H} + \frac{\omega_{y}}{H}} \right\}}}\end{matrix} & (143)\end{matrix}$where as usual data is only rendering between 0≦X≦D and 0≦Y≦R.

It is worth pointing out here that the render data for an entirecomposite hologram may well be generated at one time and then this datamay be sorted into data that is relevant to the various respectivesub-holograms. Alternatively, as we have envisaged in equation 143, datais rendered individually for each sub-hologram.

With reference to FIG. 25 a the previously derived equation u=Π/2+w isseen to remain valid as do the equations x=X and y=Y. However the ξequation changes to

$\begin{matrix}{{\xi + \frac{D - L}{2} + \omega_{x}} = {X + w}} & \left( {143a} \right)\end{matrix}$By analysis of FIG. 25 b the vertical window constraint is seen to be:

$\begin{matrix}\begin{matrix}{v \geq {\frac{\Sigma}{2} - Y + \frac{R}{2} + \omega_{y} - \frac{\Gamma}{2}}} \\{v \leq {\frac{\Sigma}{2} - Y + \frac{R}{2} + \omega_{y} + \frac{\Gamma}{2}}}\end{matrix} & \left( {143b} \right)\end{matrix}$or in terms of indices

$\begin{matrix}{v \geq {{\frac{N_{V} - 1}{\Sigma}\left\{ {\frac{\Sigma}{2} - {\frac{\left( {\beta - 1} \right)}{\left( {N_{B} - 1} \right)}R} + \frac{\left( {R - \Gamma} \right)}{2} + \omega_{y}} \right\}} + 1}} \\{v \leq {{\frac{N_{V} - 1}{\Sigma}\left\{ {\frac{\Sigma}{2} - {\frac{\left( {\beta - 1} \right)}{\left( {N_{B} - 1} \right)}R} + \frac{\left( {R + \Gamma} \right)}{2} + \omega_{y}} \right\}} + 1}}\end{matrix}$The mask transformation for a single parallax hologram with an offsetrectangular window may therefore be written as follows:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{25mu}{when}\mspace{14mu} v} \geq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R - \Gamma}{2\Sigma} + \frac{\omega_{y}}{\Sigma}} \right\}} + 1}}} & \left( {143c} \right) \\{\mspace{140mu}{{{and}\mspace{14mu} v} \leq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right)R}{\left( {N_{B} - 1} \right)\Sigma} + \frac{R + \Gamma}{2\Sigma} + \frac{\omega_{y}}{\Sigma}} \right\}} + 1}}} & \; \\{\mspace{140mu}{{{and}\mspace{14mu} 0} \leq k \leq N_{K}}} & \; \\{\mspace{59mu}{= {0\mspace{40mu}{otherwise}}}} & \left( {143d} \right) \\{where} & \; \\{{k = {{\left\{ \frac{N_{K} - 1}{L} \right\}\left\{ {\frac{L - D}{2} + \frac{\left( {\alpha - 1} \right)D}{N_{A} - 1} + \frac{\left( {\mu - 1} \right)\Pi}{N_{M} - 1} - \frac{\Pi}{2} - \omega_{x}} \right\}} + 1}},} & \left( {143e} \right) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & \left( {143f} \right) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & \left( {143g} \right)\end{matrix}$It should be noted that the horizontal printer FOV must always be largeror equal to its counterpart camera FOV in this case. The verticalprinter FOV will also need to be significantly larger than itscounterpart camera FOV (see FIG. 25 b). Since the viewing window willalways be the same for each component sub-hologram we will generallyrequire that Π≧D_(T)+L and Σ≧R_(T)+Γ where D_(T) and R_(T) arerespectively the width and height of the complete assembled hologram.

We may now trivially generalize the above example to the full parallaxcase. The mask transformation (note that we require Θ=Γ) now becomes:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} 0} < k \leq N_{K}}} & \left( {143h} \right) \\{\mspace{146mu}{{{and}\mspace{14mu} 0} < g \leq N_{G}}} & \; \\{\mspace{59mu}{= {0\mspace{45mu}{otherwise}}}} & \; \\{where} & \; \\{k = {{\left\{ \frac{N_{K} - 1}{L} \right\}\left\{ {\frac{L - D}{2} + \frac{\left( {\alpha - 1} \right)D}{N_{A} - 1} + \frac{\left( {\mu - 1} \right)\;\Pi}{N_{M} - 1} - \frac{\Pi}{2} - \omega_{x}} \right\}} + 1}} & \left( {142i} \right) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & \left( {142j} \right) \\{g = {{\left\{ \frac{N_{G} - 1}{\Gamma} \right\}\left\{ {\frac{\Gamma - R}{2} + \frac{\left( {\beta - 1} \right)R}{N_{B} - 1} + \frac{\left( {v - 1} \right)\Sigma}{N_{V} - 1} - \frac{\Sigma}{2} - \omega_{y}} \right\}} + 1}} & \left( {142k} \right) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & \left( {142l} \right)\end{matrix}$Note that in this case we require

$\begin{matrix}\begin{matrix}{\Psi_{CH} = {2\mspace{11mu}\tan^{- 1}\left\{ {\frac{D + L}{2H} + \frac{\omega_{x}}{H}} \right\}}} \\{\Psi_{CV} = {2\mspace{11mu}\tan^{- 1}\left\{ {\frac{R + \Gamma}{2H} + \frac{\omega_{y}}{H}} \right\}}}\end{matrix} & \left( {142m} \right)\end{matrix}$In addition both the horizontal and vertical printer FOVs should belarger or equal to their counterpart camera FOVs.Viewing Plane Different From Camera Plane

Up until now it has been assumed that the viewing plane is collocated atthe camera plane. However it is possible to generalize to the case wherethese two planes are not collocated. Using our intuition or moreformally, Huygens' principle, we see that in fact a full-parallaxhologram does not need to be viewed at the camera plane in order to beundistorted. In contrast, a single-parallax hologram will inevitablyshow distortion if the viewer does not view the hologram at the cameraplane. Depending on the size and depth of the single parallax hologramin question such distortion may either be important or in some cases,negligible.

In the case where digital image data is already available at one cameradistance, it may sometimes make sense to use such data for the creationof a hologram having a different viewing window position rather thanre-rendering. In this case the teaching presented hereto above willallow someone skilled in the art to derive similar equations to thosealready presented but covering the more general case of non-collocatedcamera and viewing planes.

We may also consider the possibility of defining an astigmatic viewingwindow, whose boundaries focus at different distances in the verticaland horizontal. For example the following transformation would describethe case of a single parallax hologram with an astigmatic rectangularviewing window with the horizontal window focus being located on thecamera plane at a distance H from the hologram and the vertical windowfocus being located at a distance H_(v)

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} v} \geq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right){RH}}{\left( {N_{B} - 1} \right)\Sigma\; H_{V}} + \frac{\left( {R - \Gamma} \right)H}{2\Sigma\; H_{V}}} \right\}} + 1}}} & \left( {142n} \right) \\{\mspace{135mu}{{{and}\mspace{14mu} v} \leq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - \frac{\left( {\beta - 1} \right){RH}}{\left( {N_{B} - 1} \right)\Sigma\; H_{V}} + \frac{\left( {R + \Gamma} \right)H}{2\Sigma\; H_{V}}} \right\}} + 1}}} & \; \\{\mspace{135mu}{{{and}\mspace{14mu} 0} \leq k \leq N_{K}}} & \; \\{\mspace{59mu}{= {0\mspace{34mu}{otherwise}}}} & \left( {142o} \right) \\{where} & \; \\{{k = {{\left\{ \frac{N_{K} - 1}{L} \right\}\left\{ {\frac{L - D}{2} + \frac{\left( {\alpha - 1} \right)D}{N_{A} - 1} + \frac{\left( {\mu - 1} \right)\Pi}{N_{M} - 1} - \frac{\Pi}{2}} \right\}} + 1}},} & \left( {142p} \right) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & \left( {142q} \right) \\{j = {\frac{\left( {\beta - 1} \right)\;\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & \left( {142r} \right)\end{matrix}$This transformation might be used when we wished the vertical window tohave light scrolling characteristics at the (horizontal) viewing planewith a consequently less severe vertical window collapse going away fromand towards the hologram. Such windows are particularly useful forhighly asymmetric holograms where the width, D is very different fromthe height, R.

Finally paraxial mask transformations may be derived for generalizedviewing window topologies (circles—ellipses—multiple viewing zones) andwindows possessing generalized scrolling characteristics. In additioncurved surface viewing and camera planes and the case of cameras that donot point always in the same direction are easily treated using the sameformalism as hereto described.

Objective Distortion

In any real-world optical write-head of large FOV, there is inevitablysignificant optical distortion. Usually the predominant cause of this isaberration of the objective associated with a finite 5th Seidelcoefficient. If we compare normalized object and image planes of a givenobjective as in FIG. 26 we may characterize this distortion by thefollowing transformation:

$\begin{matrix}{u = {\frac{\Pi}{2} + {\left( {U - \frac{\Pi}{2}} \right)\;{\rho\left( {U,V} \right)}}}} & (144) \\{v = {\frac{\Sigma}{2} + {\left( {V - \frac{\Sigma}{2}} \right)\;{\rho\left( {U,V} \right)}}}} & (145) \\{where} & \; \\{{\rho\left( {U,V} \right)} = {f\left( {\left( {{2U} - \Pi} \right)^{2} + \left( {{2V} - \Sigma} \right)^{2}} \right)}} & (146)\end{matrix}$and ƒ is a single-valued 1-dimensional function that describes thedistortion. These equations may also be interpreted as a transformationfrom real to paraxial object planes. By replacing the expressions

$\begin{matrix}{{u = {\left( {\mu - 1} \right)\;\frac{\Pi}{N_{M} - 1}}},\mspace{194mu}{\mu = 1},\ldots\mspace{11mu},N_{M}} & (147) \\{{v = {\left( {v - 1} \right)\;\frac{\Sigma}{N_{V} - 1}}},\mspace{205mu}{v = 1},\ldots\mspace{11mu},N_{V}} & (148) \\{with} & \; \\{{u = {\frac{\Pi}{2} + {\left\lbrack {{\left( {\mu - 1} \right)\;\frac{\Pi}{N_{M} - 1}} - \frac{\Pi}{2}} \right\rbrack\;\rho_{\mu\; v}}}},\mspace{31mu}{\mu = 1},\ldots\mspace{11mu},N_{M}} & (149) \\{{v = {\frac{\Sigma}{2} + {\left\lbrack {{\left( {v - 1} \right)\frac{\Sigma}{N_{V} - 1}} - \frac{\Sigma}{2}} \right\rbrack\;\rho_{\mu\;{v.}}}}},\mspace{45mu}{v = 1},\ldots\mspace{11mu},N_{V}} & (150)\end{matrix}$in all preceding equations we may thus derive mask-file transformationsappropriate for use in write-heads with finite distortion. First let'stake the case of the section entitled “Non-Centred Image with FixedRectangular Viewing Window of Same Size as Hologram” that treatssingle-parallax holograms. Here Π=2D Q_(X) and Q_(Y)=R. Thus

$\begin{matrix}{u = {{D + {\left\lbrack {{\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} - D} \right\rbrack\;\rho_{\mu\; v}}} = {D + w}}} & (151)\end{matrix}$The equation ξ=X+w then leads to

$\begin{matrix}{{\left( {k - 1} \right)\;\frac{D}{N_{K} - 1}} = {{\left( {\alpha - 1} \right)\;\frac{D}{N_{A} - 1}} + {\left\lbrack {{\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} - D} \right\rbrack\;\rho_{\mu\; v}}}} & (152)\end{matrix}$Similarly the equation x=D−w leads to

$\begin{matrix}{{\left( {i - 1} \right)\;\frac{2D}{N_{I} - 1}} = {D - {\left\lbrack {{\left( {\mu - 1} \right)\;\frac{2D}{N_{M} - 1}} - D} \right\rbrack\;\rho_{\mu\; v}}}} & (153)\end{matrix}$and the equation y=Y leads to

$\begin{matrix}{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (154)\end{matrix}$Likewise the vertical condition now becomes

$\begin{matrix}{{\frac{\Sigma}{2} - Y} \leq {\frac{\Sigma}{2} + {\left\lbrack {{\left( {v - 1} \right)\;\frac{\Sigma}{N_{V} - 1}} - \frac{\Sigma}{2}} \right\rbrack\;\rho_{\mu\; v}}} \leq {\frac{\Sigma}{2} - Y + {R.}}} & (155)\end{matrix}$We therefore see that equations 19-22 are replaced by the followingequations that are now valid for the finite objective distortion case:

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{{}_{}^{}{}_{}^{}}\mspace{25mu}{when}\mspace{14mu}\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma\;\rho_{\mu\; v}}}} \right\}\left( {N_{V} - 1} \right)} + 1} \leq v \leq {{\left\{ {\frac{1}{2} - {\frac{\left( {\beta - 1} \right)}{N_{B} - 1}\frac{R}{\Sigma\;\rho_{\mu\; v}}} + \frac{R}{\Sigma\;\rho_{\mu\; v}}} \right\}\left( {N_{V} - 1} \right)} + 1}}} & (156) \\{\mspace{140mu}{{{{and}\mspace{14mu} 0} < k \leq N_{K}},{{{{0 \leq i \leq N_{I}}\;\&}\mspace{14mu} 0} \leq j \leq N_{J}}}} & \; \\{\mspace{59mu}{= {0\mspace{40mu}{otherwise}}}} & \; \\{where} & \; \\{{k = {1 + \frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)} + {\left( {N_{K} - 1} \right)\;{\rho_{\mu\; v}\left\lbrack {\frac{2\left( {\mu - 1} \right)}{\left( {N_{M} - 1} \right)} - 1} \right\rbrack}}}},} & (157) \\{{i = {1 + \frac{\left( {N_{I} - 1} \right)}{2} - {\left( {N_{I} - 1} \right)\;{\rho_{\mu\; v}\left\lbrack {\frac{\left( {\mu - 1} \right)}{N_{M} - 1} - \frac{1}{2}} \right\rbrack}}}},} & (158) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (159)\end{matrix}$Note that for clarity S has now been replaced by T to distinguish datathat has been corrected for optical distortion (T) over the data S whichhas not. Equation 24 remains invariant. Note also that equations 25-27generalize to equations 157-159 and equations 31-38 are not relevant forthe finite distortion case.

We may also generalize, for the finite distortion case, the (singleparallax) equations presented in the section entitled “Centred Imagewith Fixed Rectangular Viewing Window”. Equation 49 changes to equation156. Equation 50 changes to equation 157. Equations 51 and 52 remaininvariant as does equation 53. Equation 54 changes to equation 157.Equations 55 and 56 remain invariant.

Likewise we may generalize the (single-parallax) case of the sectionentitled “Centred Image with Maximum FOV” for finite distortion.

Here we start with the modified k equation:

$\begin{matrix}{{\left( {k - 1} \right)\frac{D + T}{N_{K} - 1}} = {\frac{T}{2} + {\left( {\alpha - 1} \right)\frac{D}{N_{A} - 1}} + \frac{T}{2} + {\left\lbrack {{\left( {\mu - 1} \right)\frac{T}{N_{M} - 1}} - \frac{T}{2}} \right\rbrack\;\rho_{\mu\; v}} - \frac{T}{2}}} & (160)\end{matrix}$from which we see that

$\begin{matrix}{k = {{\frac{D}{D + T}\left( {\alpha - 1} \right)\frac{\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)}} + {\frac{T}{2}\frac{\left( {N_{K} - 1} \right)}{\left( {D + T} \right)}} + {\frac{T}{D + T}\left( {N_{K} - 1} \right)\;{\rho_{\mu\; v}\left\lbrack {\frac{\mu - 1}{N_{M} - 1} - \frac{1}{2}} \right\rbrack}} + 1}} & (161)\end{matrix}$Equations 72,74 and 75 thus remain invariant whereas equation 73transforms to equation 161. We may derive the analogue of equation 76 bynoting that now

$\begin{matrix}{{\frac{\Sigma}{2} + {\left\lbrack {{\frac{\left( {v - 1} \right)}{N_{V} - 1}\Sigma} - \frac{\Sigma}{2}} \right\rbrack\;\rho_{\mu\; v}}} \geq {\Sigma\left\{ {\frac{1}{2} - {\frac{R}{\Sigma}\frac{\left( {\beta - 1} \right)}{N_{B} - 1}} + \frac{R - \Gamma}{2\Sigma}} \right\}}} & (162)\end{matrix}$from whence we see that

$\begin{matrix}{v \geq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - {\frac{R}{\Sigma\;\rho_{\mu\; v}}\frac{\left( {\beta - 1} \right)}{\left( {N_{B} - 1} \right)}} + \frac{R - \Gamma}{2\;\Sigma\;\rho_{\mu\; v}}} \right\}} + 1.}} & (163)\end{matrix}$The other part of equation 76 follows in the same fashion whereupon wesee that this equation transforms to

$\begin{matrix}{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{25mu}{when}\mspace{14mu} v} \geq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - {\frac{R}{\Sigma\;\rho_{\mu\; v}}\frac{\left( {\beta - 1} \right)}{\left( {N_{B} - 1} \right)}} + \frac{R - \Gamma}{2\;\Sigma\;\rho_{\mu\; v}}} \right\}} + 1}}} & (164) \\{\mspace{146mu}{{{and}\mspace{14mu} v} \leq {{\left( {N_{V} - 1} \right)\left\{ {\frac{1}{2} - {\frac{R}{\Sigma\;\rho_{\mu\; v}}\frac{\left( {\beta - 1} \right)}{\left( {N_{B} - 1} \right)}} + \frac{R + \Gamma}{2\;\Sigma\;\rho_{\mu\; v}}} \right\}} + 1}}} & \; \\{\mspace{140mu}{{{{and}\mspace{14mu} 0} < k \leq N_{K}},{{{{0 \leq i \leq N_{I}}\;\&}\mspace{14mu} 0} \leq j \leq N_{J}}}} & \; \\{\mspace{59mu}{= {0\mspace{40mu}{otherwise}}}} & \; \\{where} & \; \\{k = {{\frac{D}{D + T}\left( {\alpha - 1} \right)\frac{\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)}} + {\frac{T}{2}\frac{\left( {N_{K} - 1} \right)}{\left( {D + T} \right)}} + {\frac{T}{D + T}\left( {N_{K} - 1} \right)\;{\rho_{\mu\; v}\left\lbrack {\frac{\mu - 1}{N_{M} - 1} - \frac{1}{2}} \right\rbrack}} + 1}} & (165) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (166) \\{j = {\frac{\left( {\beta - 1} \right)\left( {N_{J} - 1} \right)}{N_{B} - 1} + 1}} & (167)\end{matrix}$Equations 73-80 transform as equations 73-75.

As a further example of the generalization of the paraxial masktransformations to their finite distortion counterparts we note thatequation 87 must be replaced by

$\begin{matrix}{k = {{\left\lbrack \frac{N_{K} - 1}{L} \right\rbrack\left\lbrack {\frac{L - D}{2} + {\left( {\alpha - 1} \right)\frac{D}{N_{A} - 1}} + {\rho_{\mu\; v}\;\Pi\left\{ {\frac{\mu - 1}{N_{M} - 1} - \frac{1}{2}} \right\}}} \right\rbrack} + 1}} & (168)\end{matrix}$Thus equation 85 then changes to equation 164. Equation 87 changes toequation 168 and equations 88 and 89 remain invariant.

The full-parallax offset-window transformation likewise generalizes tothe following form:

$\begin{matrix}{{{{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}} = {{{{}_{}^{}{}_{}^{}}\mspace{20mu}{when}\mspace{14mu} 0} < k \leq N_{K}}},{0 \leq i \leq N_{I}},{0 \leq j \leq N_{J}}} & \left( {168a} \right) \\{\mspace{149mu}{{{and}\mspace{14mu} 0} < g \leq N_{G}}} & \; \\{\mspace{59mu}{= {0\mspace{50mu}{otherwise}}}} & \; \\{where} & \; \\{k = {{\left\lbrack \frac{N_{K} - 1}{L} \right\rbrack\left\lbrack {\frac{L - D}{2} + {\left( {\alpha - 1} \right)\frac{D}{N_{A} - 1}} + {\rho_{\mu\; v}\;\Pi\left\{ {\frac{\mu - 1}{N_{M} - 1} - \frac{1}{2}} \right\}} - \omega_{X}} \right\rbrack} + 1}} & \left( {168b} \right) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & \left( {168c} \right) \\{g = {{\left\lbrack \frac{N_{G} - 1}{\Gamma} \right\rbrack\left\lbrack {\frac{\Gamma - R}{2} + {\left( {\beta - 1} \right)\;\frac{R}{N_{B} - 1}} + {\rho_{\mu\; v}\Sigma\left\{ {\frac{v - 1}{N_{V} - 1} - \frac{1}{2}} \right\}} - \omega_{Y}} \right\rbrack} + 1}} & \left( {168d} \right) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & \left( {168e} \right)\end{matrix}$

Following the technique that we have described and illustrated above, itwill be clear to those skilled in the art how all single and doubleparallax paraxial mask transformations may be generalized to theirfinite distortion counterparts for the case of a static SLM.

In the case that the SLM is moved during the writing of the hologramequations 144 to 146 are modified. FIG. 27 shows a normalized SLM objectplane that is now displaced from the centre of the objective by anamount τ in the x direction and by an amount η in the y direction (seeequations 92 and 93). Equations 144 to 146 thus become

$\begin{matrix}\begin{matrix}{u = {\frac{\Pi}{2} - \tau + {\left( {U - \frac{\Pi}{2} + \tau} \right)\;{\rho\left( {{U + \tau},{V + \eta}} \right)}}}} \\{v = {\frac{\Sigma}{2} - \eta + {\left( {V - \frac{\Sigma}{2} + \eta} \right)\;{\rho\left( {{U + \tau},{V + \eta}} \right)}}}} \\{where} \\{{\rho\left( {{U + \tau},{V + \eta}} \right)} = {f\left( {\left( {{2U} - \Pi + {2\;\tau}} \right)^{2} + \left( {{2V} - \Sigma + {2\;\eta}} \right)^{2}} \right)}}\end{matrix} & \left( {169\text{-}170} \right)\end{matrix}$

We note that, as before, ƒ is a simple single-valued one dimensionalfunction. We also note that τ is a function of α and that η is afunction of β. We may thus define the tensor ρ_(αβμν) as representingthe distortion function at every point of interest.

As before, by replacing the expressions

$\begin{matrix}\begin{matrix}{{u = {\left( {\mu - 1} \right)\;\frac{\Pi}{N_{M} - 1}}},} & {{\mu = 1},\ldots\mspace{11mu},N_{M}}\end{matrix} & (171) \\\begin{matrix}{{v = {\left( {v - 1} \right)\;\frac{\Sigma}{N_{V} - 1}}},} & {{v = 1},\ldots\mspace{11mu},N_{V}}\end{matrix} & (172) \\{with} & \; \\\begin{matrix}{{u = {\frac{\Pi}{2} - \tau + {\left\lbrack {{\left( {\mu - 1} \right)\;\frac{\Pi}{N_{M} - 1}} - \frac{\Pi}{2} + \tau} \right\rbrack\;\rho_{\alpha\;\beta\;\mu\; v}}}},} & {{\mu = 1},\ldots\mspace{11mu},N_{M}}\end{matrix} & (173) \\\begin{matrix}{{v = {\frac{\Sigma}{2} - \eta + {\left\lbrack {{\left( {v - 1} \right)\;\frac{\Sigma}{N_{V} - 1}} - \frac{\Sigma}{2} + \eta} \right\rbrack\;\rho_{\alpha\;\beta\;\mu\; v}}}},} & {\mspace{11mu}{{v = 1},\ldots\mspace{11mu},N_{V}}}\end{matrix} & (174)\end{matrix}$in all preceding equations we may thus derive mask-file transformationsappropriate for the case of a moving SLM and for use in write-heads withfinite distortion. Note that τ and η are given by equations 92 and 93.

All static and moving SLM mask-file transformations presented hithertomay be converted in the above fashion to finite distortion mask-filetransforms. In practice only these finite distortion transforms are ofpractical use. One might imagine that an alternative procedure would beto employ paraxial transforms and then to reapply other transforms tocorrect for finite optical distortion.

For instance, if we define ^({hacek over (μ)}{hacek over (ν)})T_(αβ) asrepresenting the mask information corrected for finite distortion thenthe following transformation will convert the paraxial mask informationS into the required corrected form T:

$\begin{matrix}{{{}_{}^{\overset{\Cup}{\mu}\overset{\Cup}{v}}{}_{\alpha\;\beta}^{}} = {{}_{}^{\mu\; v}{}_{\alpha\;\beta}^{}}} & \left( {168a} \right) \\{where} & \; \\{\mu = {1 + {\rho_{\overset{\Cup}{\mu}\overset{\Cup}{v}}\left\{ {\overset{\Cup}{\mu} - 1 - \frac{N_{M} - 1}{2}} \right\}} + \frac{N_{M} - 1}{2}}} & \left( {168b} \right) \\{v = {1 + {\rho_{\overset{\Cup}{\mu}\overset{\Cup}{v}}\left\{ {\overset{\Cup}{v} - 1 - \frac{N_{V} - 1}{2}} \right\}} + \frac{N_{V} - 1}{2}}} & \left( {168c} \right)\end{matrix}$However, such sequential application of transforms is undesirable andwould lead to a significant increase both in computational time and mostimportantly to interpolation error.

The reason that sequential transforms act to significantly increaseinterpolation error is that usually paraxial mask-file transforms mustbe computed using a truncation type of interpolation. This is becausethe ‘magic number’ representation presented earlier is rarely flexibleenough for practical commercial use. Thus, in sequential application,the index rules for the integers i, j, k and g are truncated from arational representation to an integer representation, giving rise to atruncation error. Then later the index rules of 168b and 168c exacerbatethe error by converting a generally irrational RHS containing the priortruncation error into an integer RHS. Additionally the constraint thatthird-party 3D software programs must be used implies that specializednon-Cartesian meshes are not an option. Even if more advanced forms ofinterpolation are used in the calculation of S from I (over and abovesimple truncation methods) one inevitably incurs an interpolative error.By using sequential transforms such errors are compounded whereas byformulating a single finite distortion mask transformation only oneinterpolation error is produced. It is thus vital that a singletransform is derived that treats both the ideal paraxial mask-filetransform and the write-head optical distortion.

Generalization to Multiple Colour

All of the mask transforms above are valid for each and every colourchannel employed in the holographic printer. Note that each colourchannel will have its own distortion function and as such ρ_(μν) andρ_(αβμν) will be different for each colour.

Other Distortions

Many other image distortions may arise in a holographic printing system.Amongst these are distortions caused by emulsion swelling due tochemical processing, replay wavelength not equal to record wavelength,reference Beam angle different on replay to recording, object Beam axialangle different on replay to recording and further holographictransferring.

Many of these distortions are mathematically similar to the objectivedistortion discussed above (although they may lack that distortion'srotational symmetry properties and they may be very different for eachcolour channel). This is, because such distortions usually act only tochange the index swapping laws in the mask transformation. Thereforesuch distortions may be incorporated into a single mask transformationjust as we have discussed above for the case of objective distortion. Byderiving a single finite distortion mask transformation thatincorporates all distortions inherent in the printing system we benefitgreatly in terms of interpolation noise and computational speed.

Numerical Pre-distortion of Image Data for Compensation of Disparitybetween Recording and Replay Geometries

It has been seen above how fundamental (perspective view) image datamust be transformed in order to write 1-step holograms. In particularthe importance of including, in the definition of a single transform,all of the printing system's distortions has been underlined.

The objective distortion that has been discussed at length above iscritical to the operation of the class of printers under discussion(i.e. that are suitable for writing medium to large format holograms).It is thus impossible to “minimize” this distortion as the performanceof the printer is directly related to its magnitude. However, this isnot necessarily the case with the other subsidiary distortions that wehave introduced. The most important of these in 1-step holographicprinting devices is usually the disparity of the recording and replayreference ray geometries. This arises due to the fact that largerholograms are inevitably lit by point sources and yet the simplest andmost elegant solution for the recording is a collimated reference beam.

One approach that has been advocated in PCT applications WO 00/29909 andWO 0029908 is to essentially eliminate all distortions caused bydisparity in the recording and illumination reference ray geometriessuch that in fact no numerical compensation for this distortion isrequired. However this leads to a complex mechanical solution for thetwo dimensional control of the recording reference beam and, as we shallsee below, to non-optimum viewing characteristics of the final hologram.

We disclose here two solutions to the present problem relevant toreflection type holograms. The first uses a simple fixed and collimatedreference beam for recording. Mask transforms are then defined that takeinto account this fact and the required replay geometry (usually a pointsource at some distance). The second solution utilizes a combination ofdeliberate image pre-distortion (integrated into a mask transform asabove) and deliberate over-correction of the recording reference ray.Whilst being electromechanically more complicated this second solutionenjoys the merit of producing holograms with superior viewingcharacteristics.

Geometrical Ray Analysis of a Hologram

In order to understand how the digital image data that are used torecord a hologram may be pre-distorted such that the final holographicimage appears undistorted when viewed under a chosen lighting geometry,we will now study exactly how the holographic image is distorted whenrecording and replay geometries are different.

Model

Firstly let us use a standard right-handed spherical coordinate systemcentred on a given holopixel as depicted in FIG. 28. We define anarbitrary point on the object ray as (r_(o),θ_(o),Φ_(o)) and anarbitrary point on the reference light ray as (r_(r),θ_(r),φ_(r)). Onreplay of the holopixel we define an arbitrary point on the illuminationray as (r_(c),θ_(c),φ_(c)) and an arbitrary point on the reconstructedimage ray as (r_(i),θ_(i),φ_(i)). Next we write down theBragg-Diffraction equations for a single holopixel for reflectiongeometry:k ₂[sinφ_(i) sin θ_(i)+sinθ_(c) sinφ_(c)]_(ext) =k ₁[sinθ_(o)sinφ_(o)+sinθ_(r) sinφ_(r)]_(ext)  (175)k ₂[cosφ_(i) sinθ_(i)+sinθ_(c) cosφ_(c)]_(ext) =k ₁[sinθ_(o)cosφ_(o)+sinθ_(r) cosφ_(r)]_(ext)  (176)k ₂[cosθ_(i)+cosθ_(c)]_(int)=αk₁[cosθ_(o)+cosθ_(r)]_(int)  (177)n sinθ_(int)=sinθ_(ext)  (178)Here k₁ is the wavenumber at recording and k₂ is the wavenumber atreplay. The parameter α represents a factor describing how much theemulsion is swollen before recording. The subscript ‘ext’ refers to theθ angles just outside the emulsion layer whereas the subscript ‘int’refers to θ angles inside the emulsion. Equation 178, which is Snell'slaw, connects these two types of angles. Note that in the geometry thatwe have chosen φ is invariant across the emulsion/air interface.

Equations 175-177 may be derived in a number of ways, the simplest beingto demand that

$\begin{matrix}{\overset{->}{K} = {{{\overset{->}{k}}_{r} - {\overset{->}{k}}_{O}} = {{- {\underset{\underset{\_}{\_}}{\alpha}:\left( {{\overset{->}{k}}_{C} - {\overset{->}{k}}_{i}} \right)}} = {\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & \alpha\end{pmatrix}\left( {{\overset{->}{k}}_{C} - {\overset{->}{k}}_{i}} \right)}}}} & \left( {178a} \right)\end{matrix}$where the quantities

$\begin{matrix}\begin{matrix}\begin{matrix}{{{\overset{->}{k}}_{r} \equiv \begin{pmatrix}{k_{1}\mspace{11mu}\sin\mspace{11mu}\theta_{r}\mspace{11mu}\sin\mspace{11mu}\phi_{r}} \\{{- k_{1}}\mspace{11mu}\sin\mspace{11mu}\theta_{r}\mspace{11mu}\cos\mspace{11mu}\phi_{r}} \\{k_{1}\mspace{11mu}\cos\mspace{11mu}\theta_{r}}\end{pmatrix}},} & {{\overset{->}{k}}_{o} \equiv \begin{pmatrix}{{- k_{1}}\mspace{11mu}\sin\mspace{11mu}\theta_{O}\mspace{11mu}\sin\mspace{11mu}\phi_{O}} \\{k_{1}\mspace{11mu}\sin\mspace{11mu}\theta_{O}\mspace{11mu}\cos\mspace{11mu}\phi_{O}} \\{{- k_{1}}\mspace{11mu}\cos\mspace{11mu}\theta_{O}}\end{pmatrix}}\end{matrix} \\\begin{matrix}{{\overset{->}{k}}_{c} \equiv \begin{pmatrix}{{- k_{2}}\mspace{11mu}\sin\mspace{11mu}\theta_{C}\mspace{11mu}\sin\mspace{11mu}\phi_{C}} \\{k_{2}\mspace{11mu}\sin\mspace{11mu}\theta_{C}\mspace{11mu}\cos\mspace{11mu}\phi_{C}} \\{{- k_{2}}\mspace{11mu}\cos\mspace{11mu}\theta_{C}}\end{pmatrix}} & {and} & {{\overset{->}{k}}_{i} \equiv \begin{pmatrix}{k_{2}\mspace{11mu}\sin\mspace{11mu}\theta_{i}\mspace{11mu}\sin\mspace{11mu}\phi_{i}} \\{{- k_{2}}\mspace{11mu}\sin\mspace{11mu}\theta_{i}\mspace{11mu}\cos\mspace{11mu}\phi_{i}} \\{k_{2}\mspace{11mu}\cos\mspace{11mu}\theta_{i}}\end{pmatrix}}\end{matrix}\end{matrix} & \left( {178b} \right)\end{matrix}$are the respective wavevectors, {right arrow over (K)} is the normalfringe plane vector, α is the emulsion swelling matrix and all anglesare internal. Equations 175-177 then correspond to the x, y and zcomponents of this equation. Note that the x and y components areinvariant under internal-external angle transformations. Note also thatthe z component is independent of azimuthal coordinate. This is becausez is orthogonal to {circumflex over (φ)}.

Now let us superimpose a right-handed Cartesian system on our sphericalsystem as depicted in FIGS. 29( a-d). Note that the origin(x,y,z)=(0,0,0) coincides with the centre of the 1-step hologram. Theorigin of the spherical system, however, coincides with the holopixelunder scrutiny.

The plane (x,y,z=h_(r)) is the recording plane. The plane (x,y,z=h_(ν))is the viewing plane. The plane (x,y,z=0) is the hologram plane. Thepoint (X_(h),y_(h),0) is the holopixel of interest. The point(x₁,Y₁,h_(r)) represents the intersection of the recording object ray(that intersects with the holopixel and the actual object point) withthe recording plane. The point (x₂,y₂,h_(ν)) represents the intersectionof the replayed image ray (emanating from the holopixel) with theviewing plane.

The point (X_(c),y_(c),z_(c)) is the location of the point sourceillumination on replay. Note that z_(c)>0 and that y_(c) is negative forthe given form of equations 175 to 178. Initially we will treat the caseof a collimated reference beam in which case we will just describe thisset of rays by the spherical coordinate θ_(r). However, later on we willexamine the case where the reference beam on recording is changed duringthe process of writing. In this case the intersection of the variousrays will form a point sink at (x_(r), y_(r), z_(r)) with z_(r) positive(and y_(r) negative) (see FIG. 29 e)

Collimated Reference Wave Geometry

Now let us write down the various relations that exist betweenrepresentations in the spherical and Cartesian coordinate systems:x _(h) =x _(c) −r _(c) sinθ_(c) sinφ_(c)  (179)y _(h) =y _(c) +r _(c) sinθ_(c) cosφ_(c)  (180)x _(h) =x ₁ −r ₁ sinθ_(o) sinφ_(o)  (181)y _(h) =y ₁ +r ₁ sinθ_(o) cosφ_(o)  (182)x _(h) =x ₂ −r ₂ sinθ_(i) sinφ_(i)  (183)y _(h) =y ₂ +r ₂ sinθ_(i) cosφ_(i)  (184)

We can trivially rearrange these expressions to give equations for theterms present in equations 175 to 176:

$\begin{matrix}{{\sin\mspace{11mu}\theta_{C}\mspace{11mu}\sin\mspace{11mu}\phi_{C}} = \frac{x_{C} - x_{h}}{r_{c}}} & (185) \\{{\sin\mspace{11mu}\theta_{C}\mspace{11mu}\cos\mspace{11mu}\phi_{C}} = \frac{y_{h} - y_{C}}{r_{c}}} & (186) \\{{\sin\mspace{11mu}\theta_{O}\mspace{11mu}\sin\mspace{11mu}\phi_{O}} = \frac{x_{1} - x_{h}}{r_{1}}} & (187) \\{{\sin\mspace{11mu}\theta_{O}\mspace{11mu}\cos\mspace{11mu}\phi_{O}} = \frac{y_{h} - y_{1}}{r_{1}}} & (188) \\{{\sin\mspace{11mu}\theta_{i}\mspace{11mu}\sin\mspace{11mu}\phi_{i}} = \frac{x_{2} - x_{h}}{r_{2}}} & (189) \\{{\sin\mspace{11mu}\theta_{i}\mspace{11mu}\cos\mspace{11mu}\phi_{i}} = \frac{y_{h} - y_{2}}{r_{2}}} & (190)\end{matrix}$Now let us combine equations 177 and 178:

$\begin{matrix}{{\sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{i}}{n^{2}}} + \sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{C}}{n^{2}}}} = {\frac{\alpha\; k_{1}}{k_{2}}\left\{ {\sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{O}}{n^{2}}} + \sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{r}}{n^{2}}}} \right\}}} & (191)\end{matrix}$Expressions for sinθ₀, sinθ₁ and sinθ_(c) may be obtained from equations179 to 184. Specifically

$\begin{matrix}{{\sin^{2}\mspace{11mu}\theta_{C}} = \frac{\left( {x_{C} - x_{h}} \right)^{2} + \left( {y_{h} - y_{C}} \right)^{2}}{r_{C}^{2}}} & (192) \\{{\sin^{2}\mspace{11mu}\theta_{O}} = \frac{\left( {x_{1} - x_{h}} \right)^{2} + \left( {y_{h} - y_{1}} \right)^{2}}{r_{1}^{2}}} & (193) \\{{\sin^{2}\mspace{11mu}\theta_{i}} = \frac{\left( {x_{2} - x_{h}} \right)^{2} + \left( {y_{h} - y_{2}} \right)^{2}}{r_{2}^{2}}} & (194)\end{matrix}$Hence substituting equations 192 to 194 into 191 we obtain

$\begin{matrix}{{k_{2}\left\lbrack {\sqrt{1 - \frac{\left( {x_{2} - x_{h}} \right)^{2} + \left( {y_{2} - y_{h}} \right)^{2}}{n^{2}r_{2}^{2}}} + \sqrt{1 - \frac{\left( {x_{C} - x_{h}} \right)^{2} + \left( {y_{C} - y_{h}} \right)^{2}}{n^{2}r_{C}^{2}}}} \right\rbrack} = {k_{1}{\alpha\left\lbrack {\sqrt{1 - \frac{\left( {x_{1} - x_{h}} \right)^{2} + \left( {y_{1} - y_{h}} \right)^{2}}{n^{2}r_{1}^{2}}} + \sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{r}}{n^{2}}}} \right\rbrack}}} & (195)\end{matrix}$

Note that θ_(r) has been left explicitly in this expression as the caseof a collimated recording reference beam, characterized by zeroazimuthal angle and a given altitudinal angle θ_(r), is being discussed.We now introduce the following variablesε=x _(h) −x ₁  (196)τ=x _(h) −x ₂  (197)γ=y ₂ −y _(h)  (198)σ=y ₁ −y _(h)  (199)μ=y _(c) −y _(h)  (200)ξ=x _(h) −x _(c).  (201)Using these expressions in equations 185-190 we can rewrite equations175 and 176:

$\begin{matrix}{{k_{2}\left( {\frac{\tau}{r_{2}} + \frac{\xi}{r_{C}}} \right)} = \frac{k_{1}ɛ}{r_{1}}} & (202) \\{{k_{2}\left( {\frac{\gamma}{r_{2}} + \frac{\mu}{r_{C}}} \right)} = {k_{1}\left( {\frac{\sigma}{r_{1}} - {\sin\mspace{11mu}\theta_{r}}} \right)}} & (203)\end{matrix}$Dividing these two equations we eliminate k₁ and k₂:

$\begin{matrix}{{\left( {\frac{\tau}{r_{2}} + \frac{\xi}{r_{C}}} \right)\left( {\frac{\sigma}{r_{1}} - {\sin\mspace{11mu}\theta_{r}}} \right)} = {\left( \frac{ɛ}{r_{1}} \right)\left( {\frac{\gamma}{r_{2}} + \frac{\mu}{r_{C}}} \right)}} & (204)\end{matrix}$Now let us substitute equations 196 to 201 into 195:

$\begin{matrix}{{k_{2}\left( {\sqrt{1 - \frac{\tau^{2} + \gamma^{2}}{n^{2}r_{2}^{2}}} + \sqrt{1 - \frac{\xi^{2} + \mu^{2}}{n^{2}r_{C}^{2}}}} \right)} = {k_{1}{\alpha\left( {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}r_{1}^{2}}} + \sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{r}}{n^{2}}}} \right)}}} & (205)\end{matrix}$Using equation 202 we can now eliminate k₁ and k₂:

$\begin{matrix}{{\left( \frac{ɛ}{r_{1}} \right)\left( {\sqrt{1 - \frac{\tau^{2} + \gamma^{2}}{n^{2}r_{2}^{2}}} + \sqrt{1 - \frac{\xi^{2} + \mu^{2}}{n^{2}r_{C}^{2}}}} \right)} = {{\alpha\left( {\frac{\tau}{r_{2}} + \frac{\xi}{r_{C}}} \right)}\left( {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}r_{1}^{2}}} + \sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{r}}{n^{2}}}} \right)}} & (206)\end{matrix}$We must now write down relations describing the radial co-ordinates r₁,r₂ and r_(c). From FIGS. 29 b,c and d we see straight away that:r _(c) ²=ξ²+μ² +z _(c) ²  (207)r ₁ ²=ε²+σ² +h _(r) ²  (208)r ₂ ²=τ²+γ² +h _(v) ²  (209)The (γ,τ) Equations

Equations 204, 206, 207, 208 and 209 now constitute a complete set ofequations for the variables γ and τ. These two variables respectivelytell us the y and x co-ordinates of the intersection of the diffractedray emanating from the holopixel at (x_(h),y_(h),0) with the viewingplane, on replay of the hologram. The equations allow us to thuscalculate γ and τ if we know the geometry of the playback light and allthe recording geometry. The equations are quartic in nature and may bestbe written in parametric form as follows:

$\begin{matrix}{{{a\;\tau} + {b\;\gamma} + {g\; R}} = 0} & (210) \\{\frac{d\;\tau}{R} = {\sqrt{1 - \frac{\tau^{2} + \gamma^{2}}{n^{2}R^{2}}} + c}} & (211)\end{matrix}$

$\begin{matrix}{R^{2} = {\tau^{2} + \gamma^{2} + h_{v}^{2}}} & (212)\end{matrix}$Here we have used R=r₂ for simplicity and

$\begin{matrix}{a = {\frac{\sigma}{r_{1}} - {\sin\mspace{11mu}\theta_{r}}}} & (213) \\{b = {- \frac{ɛ}{r_{1}}}} & (214) \\{c = {\sqrt{1 - \frac{\xi^{2} + \mu^{2}}{n^{2}r_{C}^{2}}} - \frac{\xi\; d}{r_{C}}}} & (215) \\{d = {\frac{r_{1}\alpha}{ɛ}\left( {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}r_{1}^{2}}} + \sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{r}}{n^{2}}}} \right)}} & (216) \\{g = {{\frac{\xi}{r_{C}}\left( {\frac{\sigma}{r_{1}} - {\sin\mspace{11mu}\theta_{r}}} \right)} - \frac{ɛ\;\mu}{r_{1}r_{C}}}} & (217)\end{matrix}$

As we have the above equations 44-46 are quartic and hence have 4solutions. Two of these solutions have negative values of R and hence wewill disregard these as we have adopted the convention of positive R inthe definition of our spherical coordinate system in order to maintainits single-valuedness. The first solution for positive R is:

$\begin{matrix}{\tau = {\frac{h_{v}}{2\;\Omega_{4}d\; n^{2}}\sqrt{\frac{\Omega_{5}}{\Omega_{1}}}\begin{Bmatrix}{b^{2} + a^{2} + {n^{2}d^{2}b^{2}} - {g^{2}n^{2}d^{2}} - {n^{2}b^{2}} -} \\{{a^{2}n^{2}} + {n^{2}c^{2}b^{2}} + {n^{2}c^{2}a^{2}} -} \\{\frac{\Omega_{1}}{\Omega_{5}}\left( {b^{2} + a^{2} + {n^{2}d^{2}b^{2}}} \right)}\end{Bmatrix}}} & (218) \\{\gamma = {{- \frac{h_{v}}{2\;\Omega_{4}d\; n^{2}b}}\sqrt{\frac{\Omega_{5}}{\Omega_{1}}}\begin{Bmatrix}{{a\; b^{2}} + a^{3} + {a\; n^{2}d^{2}b^{2}} + {a\; g^{2}n^{2}d^{2}} -} \\{{a\; n^{2}b^{2}} - {a^{3}n^{2}} + {a\; n^{2}c^{2}b^{2}} +} \\{{n^{2}c^{2}a^{3}} + {2g\; n^{2}d\; c\; b^{2}} + {2g\; n^{2}d\; c\; a^{2}} -} \\{\frac{\Omega_{1}}{\Omega_{5}}\left( {{a\; b^{2}} + a^{3} + {a\; n^{2}d^{2}b^{2}}} \right)}\end{Bmatrix}}} & (219) \\{R = {h_{v}\sqrt{\frac{\Omega_{5}}{\Omega_{1}}}}} & (220)\end{matrix}$The second solution for positive R is:

$\begin{matrix}{\tau = {\frac{h_{v}}{2\Omega_{4}d\; n^{2}}\sqrt{\frac{\Omega_{3}}{\Omega_{1}}}\begin{Bmatrix}{b^{2} + a^{2} + {n^{2}d^{2}b^{2}} - {g^{2}n^{2}d^{2}} - {n^{2}b^{2}} -} \\{{a^{2}n^{2}} + {n^{2}c^{2}b^{2}} + {n^{2}c^{2}a^{2}} -} \\{\frac{\Omega_{1}}{\Omega_{3}}\left( {b^{2} + a^{2} + {n^{2}d^{2}b^{2}}} \right)}\end{Bmatrix}}} & (221)\end{matrix}$

$\begin{matrix}{\gamma = {{- \frac{h_{v}}{2\Omega_{4}d\; n^{2}b}}\sqrt{\frac{\Omega_{3}}{\Omega_{1}}}\begin{Bmatrix}{{a\; b^{2}} + a^{3} + {a\; n^{2}d^{2}b^{2}} + {a\; g^{2}n^{2}d^{2}} -} \\{{a\; n^{2}b^{2}} - {a^{3}n^{2}} + {a\; n^{2}c^{2}b^{2}} +} \\{{n^{2}c^{2}a^{3}} + {2g\; n^{2}d\; c\; b^{2}} + {2g\; n^{2}d\; c\; a^{2}} -} \\{\frac{\Omega_{1}}{\Omega_{3}}\left( {{a\; b^{2}} + a^{3} + {a\; n^{2}d^{2}b^{2}}} \right)}\end{Bmatrix}}} & (222) \\{R = {h_{v}\sqrt{\frac{\Omega_{3}}{\Omega_{1}}}}} & (223)\end{matrix}$The various Ω parameters referred to in the above equations are simplealgebraic functions of a,b,c,d,g and n. We do not list these functionshere for reasons of length and for the reason that their derivation willbe evident for a person skilled in the art. For example we have derivedthese functions ourselves using a commercial symbolic manipulationprogram.

Even though there are two positive R solutions to the equations 210 to212 only one is interesting. We must choose the root by requiring thatk₂ as defined by equation 203 be non-zero and positive valued.

In this section a set of algebraic equations has been derived thatdefine the x and y replay ray intersections with the viewing plane ofthe hologram. These equations assume a certain hologram recordinggeometry (we have used the case of a collimated plane recording waveabove) and a certain hologram replay geometry (generallynon-collimated).

The (γ,τ) equations will now be applied in order to analyze exactly howthe form of the hologram viewing window distorts due to the disparitybetween recording and replay geometries.

FIG. 30 shows the case of a 1-step monochromatic reflection compositehologram (λ=526.5 nm), 80 cm high by 60 cm wide and viewed from thefront, which has been written using a collimated reference beam at 56degrees angle of incidence. A perfectly paraxial objective having ahorizontal FOV of 100 degrees has been used to write image data via anSLM. The SLM has an aspect ratio of 1:0.75 H:V. Image data is written tothe SLM un-apodized (maximum FOV case) and viewing and object planes areconsidered at 130 cm in front of the hologram surface. The hologram hasundergone no emulsion shrinkage during processing and is replayed by apoint light source situated at 3 metres distance from the hologramcentre at 56 degrees angle of incidence (the reference beam on playbackis from the underneath and in front as in the diagram of FIG. 29 b).

The hologram, in FIG. 30, is depicted as the shaded rectangle 301; therectangle 302 and the rectangle 303, respectively above-left andbelow-right of the hologram, represent the peripheral limits of theideal viewing zones (at 130 cm) under collimated replay illumination forrespectively the top-left and bottom-right holopixels. The distortedzones 304 and 305 represent the peripheral limits of the actualcalculated viewing zones at 130 cm (normal) distance from the hologram,given the point source replay illumination. Clearly these latter zonesare significantly pushed away from the hologram centre and as such muchof the final total is viewing zone only carries information about asection of the hologram. As the hologram size gets bigger and theillumination light gets closer the above situation becomes critical andthere is no one viewing zone where all the hologram is viewable in itsentirety.

Whilst the viewing distance is of the order of 1.5 times the largestdimension of the hologram and the illumination light is situated at adistance of around 3 times the largest dimension of the hologram it ispossible to pre-distort the digital image data in order to obtain anundistorted hologram (when the hologram is illuminated by a pointsource). Beyond the above cited constraints it becomes impractical touse pre-distortion as the only method to make large undistortedholograms and some manipulation of the reference beam must take place(see FIG. 31 for a simulation of a 2 m×2 m hologram, written with acollimated reference beam and illuminated from a 3 m distance by a pointsource and viewed at 2 m distance—note that the viewing windows don'tintersect).

As we have previously discussed Klug et al. use exact reference beamtracking in order to circumvent the need to pre-distort the digitalimage data. This technique cites such a method as overcoming theprohibitive computational load of pre-distortion. However we have foundthat, with rapidly advancing computer power, pre-distortion is currentlycomputationally well-treatable. Further, since in certain cases,computational pre-distortion does not yield overlapping viewing windowsfor all parts of the hologram (i.e. the fact that a viewing positiondoes not exist where all the hologram image can be observed at once), itis advantageous, for such cases (typically large holograms that are tobe viewed with a point source light at close proximity), to use acomposite technique comprising both pre-distortion and some deliberatereference beam tracking on recording. Such a composite technique turnsout to have significant advantages as it acts to bring the viewingwindows of each holopixel into better alignment than when one simplytracks the reference beam such that it matches the replay source.

Pre-distortion of Data

In the case that we elect to write the hologram with a fixed collimatedbeam, the image data must be corrected for the disparity between therecording and replay geometries. We shall therefore need to know whatinformation to write on the SLM in terms of the required undistorteddata that we want to observe when the hologram is complete. Sincecurrent SLM devices are usually fabricated from a fixed and equallyspaced grid of pixels, this operation is accomplished by use of the(γ,τ) equations to calculate the required pixel swap transformations.

In the case that an SLM device is used which supports a deformable mesh,then the required distorted image data must be calculated from the givendata at viewing using the (ε,σ) equations.

The (ε,σ) Equations

Following our previous analysis equations 204, 206, 207, 208 and 209 maybe written as follows:

$\begin{matrix}{{{a\; ɛ} + {b\;\sigma} + {g\; R}} = 0} & (224) \\{\frac{d\; ɛ}{R} = {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}R^{2}}} + c}} & (225) \\{R^{2} = {ɛ^{2} + \sigma^{2} + h_{r}^{2}}} & (226)\end{matrix}$where we have used R=r₁ this time and

$\begin{matrix}{a = {\frac{\gamma}{r_{2}} + \frac{\mu}{r_{C}}}} & (227)\end{matrix}$

$\begin{matrix}{b = {{- \frac{\tau}{r_{2}}} - \frac{\xi}{r_{C}}}} & (228) \\{c = \sqrt{1 - \frac{\sin^{2}\mspace{11mu}\theta_{r}}{n^{2}}}} & (229) \\{d = \frac{\sqrt{1 - \frac{\tau^{2} + \gamma^{2}}{n^{2}r_{2}^{2}}} + \sqrt{1 - \frac{\xi^{2} + \mu^{2}}{n^{2}r_{C}^{2}}}}{\frac{\alpha\;\tau}{r_{2}} + \frac{\alpha\;\xi}{r_{C}}}} & (230) \\{g = {\left( {\frac{\tau}{r_{2}} + \frac{\xi}{r_{C}}} \right)\;\sin\mspace{11mu}\theta_{r}}} & (231)\end{matrix}$Clearly the solution of these equations is exactly similar to equations218 to 223 with γ being replaced by σ and τ being replaced by ε. If(γ,τ) represent a desired SLM image data set at the viewing distancez=h_(ν) then the solution of the above equations gives the requireddistorted dataset (σ,ε) that must be written at the recording distancez=h_(r).Recording with a Converging Reference Wave Geometry

In the case that the replay light is too close to the hologram, makingit necessary to vary the angle of the recording reference beam frompixel to pixel at production of the hologram, we must re-derive themathematical model presented above for the case of a general convergingreference wave. We will therefore now consider the case of FIG. 29 e inwhich the reference wave is redirected for each holopixel in such a wayto form a point sink at (x_(r),y_(r),z_(r)=h_(ƒ)).

Equations 179 to 184 need to be supplemented by the following relations:x _(h) =x _(r) −r _(r) sinθ_(r) sinφ_(r)  (232)y _(h) =y ^(r) +r _(r) sinθ_(r) cosφ_(r)  (233)wherer _(r) ²=(x _(h) −x _(r))²+(y _(h) −y _(r))² +z _(r) ².  (234)

Equations 192 to 194 also need to be supplemented by:

$\begin{matrix}{{\sin^{2}\mspace{11mu}\theta_{r}} = \frac{\left( {x_{r} - x_{h}} \right)^{2} + \left( {y_{h} - y_{r}} \right)^{2}}{r_{r}^{2}}} & (235)\end{matrix}$

Following 196 to 201 we now defineβ=x _(h) −x _(r)  (236)δ=y _(r) −y _(h)  (237)whence equations 232 and 233 yieldsinθ_(r) sinφ_(r) =−β/r _(r)  (238)sinθ_(r) cosφ_(r) =−δ/r _(r)  (239)

Thus equations 202 and 203 are generalized to

$\begin{matrix}{{k_{2}\left( {\frac{\tau}{r_{2}} + \frac{\xi}{r_{C}}} \right)} = {k_{1}\left( {\frac{ɛ}{r_{1}} + \frac{\beta}{r_{r}}} \right)}} & (240) \\{{k_{2}\left( {\frac{\gamma}{r_{2}} + \frac{\mu}{r_{C}}} \right)} = {k_{1}\left( {\frac{\sigma}{r_{1}} + \frac{\delta}{r_{r}}} \right)}} & (241)\end{matrix}$

Dividing these two equations we eliminate k₁ and k₂ as in 204:

$\begin{matrix}{{\left( {\frac{\gamma}{r_{2}} + \frac{\mu}{r_{C}}} \right)\left( {\frac{ɛ}{r_{1}} + \frac{\beta}{r_{r}}} \right)} = {\left( {\frac{\sigma}{r_{1}} + \frac{\delta}{r_{r}}} \right)\left( {\frac{\tau}{r_{2}} + \frac{\xi}{r_{C}}} \right)}} & (242)\end{matrix}$

Equation 205 is now generalized to:

$\begin{matrix}{{k_{2}\left( {\sqrt{1 - \frac{\tau^{2} + \gamma^{2}}{n^{2}r_{2}^{2}}} + \sqrt{1 - \frac{\xi^{2} + \mu^{2}}{n^{2}r_{C}^{2}}}} \right)} = {k_{1}{\alpha\left( {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}r_{1}^{2}}} + \sqrt{1 - \frac{\beta^{2} + \delta^{2}}{n^{2}r_{r}^{2}}}} \right)}}} & (243)\end{matrix}$and 206 to:

$\begin{matrix}{{\left( {\frac{ɛ}{r_{1}} + \frac{\beta}{r_{r}}} \right)\left( {\sqrt{1 - \frac{\tau^{2} + \gamma^{2}}{n^{2}r_{2}^{2}}} + \sqrt{1 - \frac{\xi^{2} + \mu^{2}}{n^{2}r_{C}^{2}}}} \right)} = {{\alpha\left( {\frac{\tau}{r_{2}} + \frac{\xi}{r_{C}}} \right)}\left( {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}r_{1}^{2}}} + \sqrt{1 - \frac{\beta^{2} + \delta^{2}}{n^{2}r_{r}^{2}}}} \right)}} & (244)\end{matrix}$

Finally with the additional relationr _(r) ²=β²+δ² +h _(ƒ) ²  (245)we can recast our problem in exactly the form of equations 210 to 212.

The Generalized (γ,τ) Equations

-   -   As before we may now write        ατ+bγ+gR=0  (246)

$\begin{matrix}{\frac{d\;\tau}{R} = {\sqrt{1 - \frac{\tau^{2} + \gamma^{2}}{n^{2}R^{2}}} + c}} & (247) \\{R^{2} = {\tau^{2} + \gamma^{2} + h_{v}^{2}}} & (248)\end{matrix}$where, as before, we have used R=r₂ for simplicity. We write thegeneralized coefficient set as:

$\begin{matrix}{a = {\frac{\sigma}{r_{1}} - \frac{\delta}{r_{r}}}} & (249) \\{b = {{- \frac{ɛ}{r_{1}}} - \frac{\beta}{r_{r}}}} & (250) \\{c = {\sqrt{1 - \frac{\xi^{2} + \mu^{2}}{n^{2}r_{C}^{2}}} - \frac{\xi\; d}{r_{C}}}} & (251) \\{d = {\frac{r_{1}\alpha}{\frac{ɛ}{r_{1}} + \frac{\beta}{r_{r}}}\left( {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}r_{1}^{2}}} + \sqrt{1 - \frac{\beta^{2} + \delta^{2}}{n^{2}r_{r}^{2}}}} \right)}} & (252) \\{g = {{\frac{\xi}{r_{C}}\left( {\frac{\sigma}{r_{1}} + \frac{\delta}{r_{r}}} \right)} - {\frac{\mu}{r_{C}}\left( {\frac{ɛ}{r_{1}} + \frac{\beta}{r_{r}}} \right)}}} & (253)\end{matrix}$

These equations may then be solved using the expressions of equations218 to 223. In FIG. 32 we re-examine the problem already treated in FIG.30, but with a converging reference geometry. FIG. 32 a shows the casewhere the replay geometry is designed to be exactly conjugate to therecording geometry. In this case we see that we obtain un-distortedrectangular viewing windows as would be expected. This is clearly ratherbetter than the previous situation with a collimated reference wavegeometry. However, we still observe that in most of the viewing zoneonly part of the hologram is visible. FIG. 32 b shows a numericalsimulation for the case of an over-corrected converging referencegeometry (corresponding to the reference sink being at 1.8 m from thecentre point of the hologram). Clearly the viewing zones are nowsignificantly pushed into one another and horizontally we gain around20% on the usable FOV of the objective. Vertically the effect is smallerdue to the inclination of the replay and reference beams in this plane.Accordingly we might suppose that an astigmatic converging referencewith different rates of convergence horizontally and vertically mightalso improve the vertical alignment of the viewing windows. Such ahypothesis is tested in FIG. 32 c which shows the same case as FIG. 32 bbut with a moderately astigmatic converging reference. Although we see asignificant improvement in vertical registration we also observe aslight degradation in resolution which may become problematic in certaingeometries. Nevertheless we see, by overcorrecting the reference onrecording, using either a slightly astigmatic or a non-astigmaticgeometry, we can achieve a highly optimal viewing configuration whichhas a significant advantage over the solution of Klug et al. who desireonly to minimize image distortion.

Calculating the Best Choice for the Recording Geometry

In the case of a hologram designed to be lit with a close point source,the above section demonstrated clearly the advantage of using anovercorrected reference geometry on recording. This may be thought of asinducing an effectively benign distortion into our hologram printingsystem in order to achieve a better viewing situation.

There are various ways of varying the reference beam recording angle inorder to achieve better viewing window overlap for a certain replaygeometry.

The simplest method is to use an overcorrected reference recording beamof axial angle of incidence equal to that of the intended replay beambut which is characterized by a sink at (x_(r),y_(r),z_(r)=h_(ƒ)). Thenwe calculate h_(f) by preferably minimizing the quantity

$\begin{matrix}{P = {\sum\limits_{\alpha \neq \beta}{\sum{\frac{w\left( {\alpha,\beta} \right)}{\Lambda_{o}}{\int_{0}^{\Lambda_{o}}{\left( {\left\lbrack {{x_{2\;\alpha}(\Lambda)} - {x_{2\;\beta}(\Lambda)}} \right\rbrack^{2} + \left\lbrack {{y_{2\;\alpha}(\Lambda)} - {y_{2\;\beta}(\Lambda)}} \right\rbrack^{2}} \right)\;{\mathbb{d}\Lambda}}}}}}} & (254)\end{matrix}$Here, Λ (0≦Λ≦Λ_(o))is a coordinate that defines the location of a pointon the calculated viewing window perimeter corresponding to a certainholopixel. Its magnitude may either refer to the distance between achosen reference point, then measuring around the perimeter of theviewing window, to the point in question or it may more preferably referto the distance around the object window perimeter to the correspondingpoint (this is consistent as rays transform in a 1 to 1 manor from theobject geometry to the image geometry and hence a point on the viewingwindow perimeter is uniquely associated with a point on the objectwindow perimeter—FIG. 33). The parameters α and β determine respectivelytwo general holopixels, whose viewing windows we wish to compare. Thesum may be carried out over all α and β or only over a selecteddistributed subset which, in one limit, would be two diagonally extremepixels. The function w is a simple weighting function that is generallychosen such as to give more weight to distant holopixels than to closeneighbors. At each iteration of the numerical minimization theparameters x₂ and y₂ are calculated using the generalized (γ,τ)equations with an initial window shape (which is usually a rectangletracing the edge of the SLM). After the minimization converges h_(f)will have been determined and we may simply use the generalized (γ,τ)equations (or in some cases a generalized version of the (ε,σ)equations—e.g. for deformable mesh SLMs) to calculate the requiredpre-distorted digital data at (x_(1α), y_(1α)∀α) from the undistortedimage data at (x_(2α), y_(2α)∀α).The Generalized (ε,σ) Equations

For completeness we note that the generalized (ε,σ) equations may bewritten in canonical form as follows:

$\begin{matrix}{{{a\; ɛ} + {b\;\sigma} + {g\; R}} = 0} & (255) \\{\frac{d\; ɛ}{R} = {\sqrt{1 - \frac{ɛ^{2} + \sigma^{2}}{n^{2}R^{2}}} + c}} & (256) \\{R^{2} = {ɛ^{2} + \sigma^{2} + h_{r}^{2}}} & (257)\end{matrix}$where R=r₁. The generalized coefficients are calculated exactly asabove.Refinement of the Recording Geometry Choice

We mentioned in the previous section that it is possible to obtainsomewhat better viewing window overlap using astigmatic recording beams.In fact we may generalize this to minimizing the function P of equation254 with respect to individual altitudinal (θ_(r)) and azimuthal (φ_(r))reference recording angles corresponding to each holopixel. Equations238 and 239 may then be used in equations 246 to 248 (or 255 to 257) inorder to calculate to required pre-distorted image data.

It should be noted that for the purposes of numerical convergence thefollowing definition of P is, in some cases, preferred:

$\begin{matrix}{P = {\sum\limits_{\alpha \neq \beta}{\sum{\frac{w\left( {\alpha,\beta} \right)}{\Lambda_{o}}{\int_{0}^{\Lambda_{o}}{\left\{ {\begin{bmatrix}\left( {{x_{2\alpha}(\Lambda)} - {x_{2\beta}(\Lambda)}} \right) \\{- {\chi\left( {x_{h\;\alpha} - x_{h\;\beta}} \right)}}\end{bmatrix}^{2} + \begin{bmatrix}\left( {{y_{2\alpha}(\Lambda)} - {y_{2\beta}(\Lambda)}} \right) \\{- {\chi\left( {y_{h\;\alpha} - y_{h\;\beta}} \right)}}\end{bmatrix}^{2}} \right\}\ {\mathbb{d}\Lambda}}}}}}} & (258)\end{matrix}$When the parameter χ=0, this equation is the same as equation 254. When,however, χ=1, however, the minimum of the function P will correspond tothe closest situation possible to when all viewing windows are centredon their respective holopixel. By initially choosing χ=1 and then byslowly changing χ towards 0, we can find the minimum of P(χ=0) moreeasily. In addition, sometimes this minimum does not exist and we maythen, using this procedure find a minimum P for a finite and acceptablevalue of χ.

Various definitions of P may be constructed that are somewhat differentto those given above but which serve the same purpose. We do not givefurther examples here as it will be clear to someone skilled in the arthow many types of P function may be formulated (given the knownillumination geometry for the hologram) that result in theidentification of a specific recording geometry corresponding to optimumhologram viewing characteristics.

Indeed the problem may be cast using various other mathematicalformalisms. Each of these formalisms will seek to vary the recording ofsome or all of the reference angle parameters in order to find a bestset of such angles corresponding to the viewing windows of all (or somerepresentative set of) holopixels being generally overlapped better thanthey otherwise would.

In summary the techniques of Klug et al. search to avoid the necessityof pre-distortion of the image data by exact conjugate matching ofrecording and replay reference beams. We find that, particularly for thecase of large format holograms that are to be replayed with a pointsource light at close proximity, there is significant motivation forchoosing a combination of pre-distortion and reference beam tracking inorder to produce better viewing window overlap. This technique may beexpected to produce better quality holograms.

In the case of medium format holograms, where the replay light is not soclose to the hologram, we find that the mechanically simple solution ofa collimated recording beam combined with image pre-distortion providesan optimal solution to the problem.

The Integration of Pixel Swapping Transformations

It has been seen in the first part of this invention how digital imagedata must be transformed according to special mask transformations inorder that this data be in a form in which it can be written to the SLMof the holographic printer. We have also seen how the fundamentaldistortion of the printer writing objective has to be integrallyincorporated in such mask transformations.

In the second part of this invention we have seen how the digital datathat we intend to write onto the printer SLM must also be transformed inorder to eliminate both the various diffractive distortions inherent tothe printer (i.e. recording reference geometry) and by the viewingconditions of the final hologram (i.e. replay reference geometry).

Just as the objective distortion compensation must be integrated intothe mask transformation so too must we integrate the diffractivedistortion compensation. To see how this works it is best to formallyre-discretize the SLM vertical and horizontal co-ordinates in terms ofthe indices {hacek over (ε)} and {hacek over (τ)} just as we havepreviously used μ and ν. The reason for this is that we will then retainν and ν for the description of the projected SLM plane at viewing time.

We now define the tensor ^({hacek over (ε)}{hacek over (τ)})W_(αβ) whichrepresents the diffraction compensated data that we intend to write toour SLM for each holopixel (α,β) assuming a paraxial objective. Thistensor is related to the paraxial mask tensor S by the followingrelationship:^({hacek over (ε)}{hacek over (σ)})Wαβ=^(μν)S_(αβ)  (259)Since μ and ν correspond respectively to τ and γ (by a simple linearscaling) we can use equations 218 to 223 in order to writeμ=F({hacek over (σ)},{hacek over (ε)})  (260)ν=G({hacek over (σ)},{hacek over (ε)})  (261)where F is the function of equation 218/221 and G is the function of219/222. Note also that for the generalized case we may use thegeneralized (γ,τ) equations (this will just change the forms of F and Gbut otherwise 260 & 261 hold).

Taking the full-parallax max-FOV mask transform of equations 132 to 136as an example, we may thus write:

$\begin{matrix}{{{{}_{}^{\overset{\Cup}{\delta}\overset{\Cup}{\sigma}}{}_{}^{}} = {{{}_{}^{\mu\; v}{}_{}^{}} = {{}_{}^{}{}_{}^{}}}}{where}} & (262) \\{{k = {{\left\lbrack \frac{D}{D + T} \right\rbrack\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)}} + {\left\lbrack \frac{T}{D + T} \right\rbrack\frac{\left( {{F\left( {\overset{\Cup}{\sigma},\overset{\Cup}{ɛ}} \right)} - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{M} - 1} \right)}} + 1}},} & (263) \\{{i = {\frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} + 1}},} & (264) \\{g = {{\left\lbrack \frac{R}{R + \Sigma} \right\rbrack\frac{\left( {\beta - 1} \right)\left( {N_{G} - 1} \right)}{N_{B} - 1}} + {\left\lbrack \frac{\Sigma}{R + \Sigma} \right\rbrack\frac{\left( {{G\left( {\overset{\Cup}{\sigma},\overset{\Cup}{ɛ}} \right)} - 1} \right)\left( {N_{G} - 1} \right)}{N_{V} - 1}} + 1}} & (265) \\{j = {\frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} + 1}} & (266)\end{matrix}$

Now W must be corrected for objective distortion. If we denote T as thefully corrected mask data then it follows that

$\begin{matrix}{{{{}_{}^{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}{}_{}^{}} = {{}_{}^{\overset{\Cup}{\delta}\overset{\Cup}{\sigma}}{}_{}^{}}}{where}} & \left( {266a} \right) \\{\overset{\Cup}{ɛ} = {1 + {\rho_{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}\left\{ {\overset{\Cup}{\mu} - 1 - \frac{N_{M} - 1}{2}} \right\}} + \frac{N_{M} - 1}{2}}} & \left( {266b} \right) \\{\overset{\Cup}{\sigma} = {1 + {\rho_{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}\left\{ {\overset{\Cup}{v} - 1 - \frac{N_{V} - 1}{2}} \right\}} + \frac{N_{V} - 1}{2}}} & \left( {266c} \right)\end{matrix}$whence the integrated transform may be written:

$\begin{matrix}{{{{}_{}^{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}{}_{}^{}} = {{}_{}^{}{}_{}^{}}}{where}} & \left( {266d} \right) \\{k = {1 + {{int}\begin{Bmatrix}{{\left\lbrack \frac{D}{D + T} \right\rbrack\frac{\left( {\alpha - 1} \right)\left( {N_{K} - 1} \right)}{\left( {N_{A} - 1} \right)}} + {{\left\lbrack \frac{T}{D + T} \right\rbrack\left\lbrack \frac{N_{K} - 1}{N_{M} - 1} \right\rbrack}*}} \\\left( {F\left( {{1 + {\rho_{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}\left( {\overset{\Cup}{v} - 1 - \frac{N_{V} - 1}{2}} \right)} + \frac{N_{V} - 1}{2}},} \right.} \right. \\\left. {\left. {1 + {\rho_{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}\left( {\overset{\Cup}{u} - 1 - \frac{N_{M} - 1}{2}} \right)} + \frac{N_{M} - 1}{2}} \right) - 1} \right)\end{Bmatrix}}}} & \left( {266e} \right) \\{i = {{{int}\left\{ \frac{\left( {\alpha - 1} \right)\left( {N_{I} - 1} \right)}{N_{A} - 1} \right\}} + 1}} & \left( {266f} \right)\end{matrix}$

$\begin{matrix}{g = {1 + {{int}\begin{Bmatrix}{{\left\lbrack \frac{R}{R + \Sigma} \right\rbrack\frac{\left( {\beta - 1} \right)\left( {N_{G} - 1} \right)}{N_{B} - 1}} + {{\left\lbrack \frac{\Sigma}{R + \Sigma} \right\rbrack\left\lbrack \frac{N_{G} - 1}{N_{V} - 1} \right\rbrack}*}} \\\left( {G\left( {{1 + {\rho_{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}\left( {\overset{\Cup}{v} - 1 - \frac{N_{V} - 1}{2}} \right)} + \frac{N_{V} - 1}{2}},} \right.} \right. \\\left. {\left. {1 + {\rho_{\overset{\Cup}{\mu}\;\overset{\Cup}{v}}\left( {\overset{\Cup}{u} - 1 - \frac{N_{M} - 1}{2}} \right)} + \frac{N_{M} - 1}{2}} \right) - 1} \right)\end{Bmatrix}}}} & \left( {266g} \right) \\{j = {{{int}\left\{ \frac{\left( {N_{J} - 1} \right)\left( {\beta - 1} \right)}{N_{B} - 1} \right\}} + 1}} & \left( {266h} \right)\end{matrix}$

This in then an example of a simple mask transformation that correctsfor both diffractive and objective distortion effects. It is clearly asimple matter to apply this logic to any of the previously treatedviewing and rendering geometries in order to arrive at other masktransformations incorporating both diffractive disparity and objectivedistortion effects.

In summary, for large format reflection holograms that are designed tobe illuminated by a point source at relatively close proximity thefollowing procedure should be used. A rectangular boundary to therecording SLM is defined by a series of (ε,σ) coordinates. The functionP or equations 254/258 is then minimized, thus defining the preferredrecording geometry in terms of the required illumination geometry (i.e.this defines F and G). Then the modified mask transformation (e.g.equations 266d-266h or similar equations describing an integrated masktransformation for finite diffractive disparity and finite objectivedistortion) is applied to the digital image data in order to calculatethe required SLM data. Finally the hologram is written using a recordingreference beam tracking that is defined by the calculated optimumrecording geometry.

In the case of smaller format holograms where a collimated reference isused on recording, the formalism of transformation 266d-266h is stillvalid.

The General Case of a Precise Diffractive/Refractive Model

When we study the diffractive and refractive processes in detail thatoccur at a given holopixel, we see that equations 175 to 178 maygenerally be replaced by more general expressions of the formA ₁(θ_(i),θ_(c),θ_(o),θ_(r),φ_(i),φ_(c),φ_(o),φ_(r),k₁,k₂)=B₁  (267)A ₂(θ_(i),θ_(c),θ_(o),θ_(r),φ_(i),φ_(c),φ_(o),φ_(r),k₁,k₂)=B₂  (268)A ₃(θ_(i),θ_(c),θ_(o),θ_(r),φ_(i),φ_(c),φ_(o),φ_(r),k₁, k₂)=B₃  (269)where the functions A₁, A₂ and A₃ are general non-linear functions ofthe shown variables and the functions B₁, B₂ and B₃ are non-linearfunctions describing the refractive and emulsion-swelling processes.This leads simply to the functions F and G of equations 260 and 261being replaced by general non-linear functions. Otherwise the procedurethat we have described above remains valid.

We have illustrated the formulation of the generalized masktransformation with the max-FOV case. It should be clear to someoneskilled in the art that generally we might optimize this process byvariably clipping the final SLM data window such that each and everyviewing window corresponding to a given holopixel precisely aligned atreplay. These clipping functions are clearly defined after we haveminimized the function P.

In addition it should be evident for someone skilled in the art how allthe above procedures may be generalized to all the various geometriesthat have been hereto above discussed.

Discoloration

In the above discussions we have assumed that the swelling parameter αis known and that its effect is compensated for routinely. However, wheneither swelling is present or the replay geometry of the hologram is notequal to the recording geometry we observe that generally the replaywavelength of a given holopixel is not the same as the recordingwavelength.

In the general case of pre-distorted data that we have discussed above,we observe that for each holopixel there is imposed a slightly differentdiscoloration by the applied pre-distortion. This means that the simpleapplication of pre-distortion as described above will lead to a hologramimage that appears discolored to different extents depending onholopixel location and on viewing angle.

The resolution to this effect is to calculate the replay wavelength as afunction of holopixel location and viewing location and to use thisinformation to modify our colour mixing at each such datum. Generally,for an RGB model, we may thus write^({hacek over (μ)}{hacek over (ν)}) T _(αβ) =^({hacek over (ε)}{hacek over (σ)}) W _(αβ)=^(μν) S_(αβ)=^({hacek over (μ)}{hacek over (ν)}) U _(Gαβ) ^(kg) I_(ij)+^({hacek over (μ)}{hacek over (ν)}) U _(Rαβ) ^(kg) J_(ij)+^({hacek over (μ)}{hacek over (ν)}) U _(Bαβ) ^(kg) K _(ij)  (270)where now T is the green mask data, I is the green image data, J is thered image data, K is the blue image data and U_(G), U_(R), and U_(B) arerespectively functions of the holopixel replay wavelength and the red,green and blue recording wavelengths. Clearly the equations for the blueand red mask data are equivalent.

In some cases the wavelength shift due to the disparity betweenrecording and replay reference beam geometries may become too great foreasy correction. In this case the P functions of equations 254 and 258should be generalized to contain a term proportional to the magnitude ofthe wavelength shift. Lagrange multipliers may be used to control theimportance of these terms within the minimization. Such a constrainedoptimization will lead to the best window overlap which doesn't undulydiscolour the image.

The P functions of equations 254 and 258 may also be modified to containa term inversely proportional to the final spatial image resolution onthe viewing plane. Again Lagrange multipliers may be used to control theimportance of this term. Such a constrained optimization would assurethe identification of the best window overlap whilst producing ahologram of acceptable peripheral angular resolution.

1. A method of writing a composite 1-step hologram, comprising:generating a laser beam; acquiring digital data from an object, saiddigital data being described by a luminous intensity tensor ^(kg)I_(ij)wherein i and j are the horizontal and vertical pixel coordinates of agiven perspective view that is generated by a real or virtual camerawhose location is described by k in the horizontal dimension and g inthe vertical dimension; performing a single mathematical transformationto convert said luminous intensity tensor ^(kg)I_(ij) into a tensor^(μν)T_(αβ) wherein α and β are the horizontal and vertical coordinatesof a holographic pixel on the composite hologram and μ and ν are thehorizontal and vertical coordinates of a given pixel on a spatial lightmodulator on to which the data for each holographic pixel is written,wherein said single mathematical transformation transforms said digitaldata whilst integrally correcting said digital data for a finitedistortion of an optical objective; writing corrected data to a spatiallight modulator, wherein said corrected data is described by said tensor^(μν)T_(αβ); directing said laser beam on to said spatial lightmodulator so that at least a portion of said laser beam is spatiallymodulated by said spatial light modulator to form a spatially modulatedlaser beam; passing said spatially modulated laser beam through anoptical objective having a finite distortion, said optical objectivefocusing said spatially modulated laser beam on to a photosensitivesubstrate; directing a reference recording beam on to saidphotosensitive substrate; and forming a double-parallax compositehologram on said photosensitive substrate.
 2. A method as claimed inclaim 1, wherein said digital data is acquired from a real object andcomprises a plurality of perspective views of the required hologramimage.
 3. A method as claimed in claim 1, wherein said digital data isacquired from a virtual object and comprises a plurality of perspectiveviews of the required hologram image.
 4. A method as claimed in claim 1,wherein said composite 1-step hologram is selected from the groupconsisting of: (i) a transmission hologram; and (ii) a reflectionhologram.
 5. A method as claimed in claim 1, wherein said singlemathematical transformation generates a rectangular viewing windowlocated in front of said hologram.
 6. A method as claimed in claim 5,wherein said spatial light modulator is either: (i) static whilstwriting said hologram; or (ii) moving whilst writing said hologram.
 7. Amethod as claimed in claim 5, wherein said viewing window is either: (i)of substantially similar size to said composite hologram; or (ii) ofdifferent size to said composite hologram.
 8. A method as claimed inclaim 5, wherein said viewing window is either: (i) symmetricallylocated in front of said composite hologram; or (ii) generally offsetfrom the centre of said composite hologram.
 9. A method as claimed inclaim 5, wherein either: (i) said viewing window is located at the sameperpendicular distance from a given point in the holographic image asthe camera plane is located from the corresponding point on the objectfrom which said digital data is acquired; or (ii) said viewing window islocated at a certain perpendicular distance from a given point in theholographic image and the camera plane is located at a substantiallydifferent perpendicular distance from the corresponding point on theobject from which said digital data is acquired.
 10. A method as claimedin claim 1, wherein said digital data is generated by a real or virtualcamera which generates either: (i) a plurality of apodized images whichare centred in a frame which corresponds with the object which is to bereproduced by said hologram; or (ii) a plurality of non-apodized imageshaving frames which correspond with the object which is to be reproducedby said hologram, said frames being generally off-centred.
 11. A methodas claimed in claim 1, wherein said single mathematical transformationgenerates a scrolling viewing window located in front of said hologram.12. A method as claimed in claim 11, wherein either: (i) the cameraplane is located at a certain distance from a point on the object andthe viewing plane is located at substantially the same distance from acorresponding point in the holographic image; or (ii) the camera planeis located at a certain distance from a point on the object and theviewing plane is located at substantially a different distance from acorresponding point in the holographic image.
 13. A method as claimed inclaim 1, wherein said single mathematical transformation generates aviewing window having a fixed size in the horizontal dimension and whichscrolls in the vertical dimension, said viewing window being located infront of the hologram.
 14. A method as claimed in claim 1, wherein saidsingle mathematical transformation generates a viewing window having afixed size in the vertical dimension and which scrolls in the horizontaldimension, said viewing window being located in front of the hologram.15. A method as claimed in claim 1, wherein said composite hologram isformed using a variable angle reference recording beam.
 16. A method asclaimed in claim 15, wherein said composite 1-step hologram is replayedusing a point-source light.
 17. A method as claimed in claim 15, whereinsaid composite 1-step hologram is replayed using collimated light.
 18. Amethod as claimed in claim 1, wherein said composite hologram is formedusing a fixed angle or collimated reference recording beam.
 19. A methodas claimed in claim 1, wherein said single mathematical transformationadditionally integrally corrects said digital data for the imagedistortion caused by the altitudinal and azimuthal reference beamangle(s) used to replay each holographic pixel of said hologram beingdifferent from the altitudinal and azimuthal reference beam angle(s)used to write each holographic pixel of said hologram.
 20. A method asclaimed in claim 1, wherein said single mathematical transformationadditionally integrally pre-distorts said digital data so that the datawritten on to said spatial light modulator is distorted.
 21. A method asclaimed in claim 20, further comprising overcorrecting the referencerecording beam using either an astigmatic or a non-astigmatic geometry.22. A method as claimed in claim 21, wherein the overlap of viewingwindows of a plurality of holographic pixels is arranged to bemaximised.
 23. A method as claimed in claim 22, wherein either: (i) theangular resolution within the overall viewing window of said hologramaveraged over said overall viewing window is maximised; or (ii) theangular resolution at the periphery of said overall viewing window ismaximised.
 24. A method as claimed in claim 21, wherein the overlap ofviewing windows of two diagonally opposed holographic pixels ismaximised.
 25. A method as claimed in claim 20, further comprisingdetermining individual altitudinal and azimuthal reference recordingangles for at least a majority of the holographic pixels forming saidhologram.
 26. A method as claimed in claim 21, wherein thepre-distortion of said digital data and the step of overcorrecting thereference recording beam are arranged such that chromatic discolorationis minimized.
 27. A method as claimed in claim 1, wherein said singlemathematical transformation additionally integrally corrects saiddigital data for the distortion caused by emulsion swelling of saidsubstrate.
 28. A method as claimed in claim 1, wherein said singlemathematical transformation additionally integrally corrects saiddigital data for the distortion caused by the wavelength of light usedto replay said hologram being different from the wavelength of lightused to write said hologram.
 29. A method as claimed in claim 1, furthercomprising a plurality of colour channels.
 30. A method as claimed inclaim 29, further comprising a red and/or green and/or blue colourchannel.
 31. A method as claimed in claim 30, wherein a spatial lightmodulator is provided for each colour channel.
 32. A method as claimedin claim 29, wherein said composite 1-step hologram is a multiple colourhologram.
 33. A method as claimed in claim 32, wherein said multiplecolour hologram is formed using reference recording beams having a firstgeometry and said hologram is replayed with light rays having a geometrydifferent to said first geometry.
 34. A method as claimed in claim 33,further comprising calculating the replay wavelength as a function ofaltitudinal and azimuthal angles for at least a majority of theholographic pixels forming said hologram.
 35. A method as claimed inclaim 29, further comprising calculating linear chromatic couplingtensors for each colour channel.
 36. A method as claimed in claim 35,wherein a separate tensor ^(μν)T_(αβ) is calculated for each colourchannel.
 37. A method as claimed in claim 36, wherein a corrected tensoris calculated for each colour channel as a linear combination of eachuncorrected component colour tensor ^(μν)T_(αβ) each operated on by asaid chromatic coupling tensor.
 38. A method as claimed in claim 37,wherein for each holographic pixel each said corrected tensor is writtento a separate spatial light modulator in such a way as to create a fullycolour-corrected composite colour hologram.
 39. A method as claimed inclaim 1, wherein said single mathematical transformation between thetensors ^(kg)I_(ij) and ^(μν)T_(αβ) consists of a reordering of theelements according to a set of single index laws of the form k=ƒ₁(α, β,μ, ν, P_(l), Q_(l), H_(l), λ), g=ƒ₂(α, β, μ, ν, P_(l), Q_(l), H_(l), λ),i=ƒ₃(α, β, μ, ν, P_(l), Q_(l), H_(l), λ) and j=ƒ₄(α, β, μ, ν, P_(l),Q_(l), H_(l), λ), wherein the functions ƒ_(n) are general functions ofthe indicated indices, P_(l) are a set of parameters characterizing thephysical characteristics of the hologram, Q_(l) are a set of parameterscharacterizing the optical properties of the hologram writing mechanism,H_(l) is a set of parameters characterizing the geometrical propertiesof the reference recording and reference replay beams and λ is thewavelength at which the hologram is recorded.
 40. A 1-step holographicprinter, comprising: a laser source; control means for acquiring digitaldata from an object, said digital data being described by a luminousintensity tensor ^(kg)I_(ij) wherein i and j are the horizontal andvertical pixel coordinates of a given perspective view that is generatedby a real or virtual camera whose location is described by k in thehorizontal dimension and g in the vertical dimension, said control meansperforming a single mathematical transformation to convert said luminousintensity tensor ^(kg)I_(ij) into a tensor ^(μν)T_(αβ) wherein α and βare the horizontal and vertical coordinates of a holographic pixel onthe composite hologram and μ and ν are the horizontal and verticalcoordinates of a given pixel on a spatial light modulator on to whichthe data for each holographic pixel is written, wherein said singlemathematical transformation transforms said digital data whilstintegrally correcting said digital data for a finite objectivedistortion of an optical objective; a spatial light modulator onto whichdata described by said tensor ^(μν)T_(αβ) is written in use, wherein inuse a laser beam is directed on to said spatial light modulator so thatat least a portion of the beam profile of said laser beam is spatiallymodulated by said spatial light modulator to form a spatially modulatedlaser beam; an optical objective through which said spatially modulatedlaser beam is passed in use, said optical objective focusing in use saidspatially modulated laser beam on to a photosensitive substrate so thata double-parallax composite hologram is formed in use on to aphotosensitive substrate.
 41. A 1-step holographic printer, comprising:a laser source; control means for acquiring digital data from an object,said digital data being described by a luminous intensity tensor^(k)I_(ij) wherein i and j are the horizontal and vertical pixelcoordinates of a given perspective view that is generated by a real orvirtual camera whose location is described by k in the horizontaldimension, said control means performing a single mathematicaltransformation to convert said luminous intensity tensor ^(k)I_(ij) intoa tensor ^(μν)T_(αβ) wherein α and β are the horizontal and verticalcoordinates of a holographic pixel on the composite hologram and μ and νare the horizontal and vertical coordinates of a given pixel on aspatial light modulator on to which the data for each holographic pixelis written, wherein said single mathematical transformation transformssaid digital data whilst integrally correcting said digital data for afinite objective distortion of an optical objective; a spatial lightmodulator onto which data described by said tensor ^(μν)T_(αβ) iswritten in use, wherein in use a laser beam is directed on to saidspatial light modulator so that at least a portion of the beam profileof said laser beam is spatially modulated by said spatial lightmodulator to form a spatially modulated laser beam; an optical objectivethrough which said spatially modulated laser beam is passed in use, saidoptical objective focusing in use said spatially modulated laser beam onto a photosensitive substrate so that a single-parallax compositehologram is formed in use on to a photosensitive substrate.
 42. A methodof writing a composite 1-step hologram, comprising: generating a laserbeam; acquiring digital data from an object, said digital data beingdescribed by a luminous intensity tensor ^(k)I_(ij) wherein i and j arethe horizontal and vertical pixel coordinates of a given perspectiveview that is generated by a real or virtual camera whose location isdescribed by k in the horizontal dimension; performing a singlemathematical transformation to convert said luminous intensity tensor^(k)I_(ij) into a tensor ^(μν)T_(αβ) wherein α and β are the horizontaland vertical coordinates of a holographic pixel on the compositehologram and μ and ν are the horizontal and vertical coordinates of agiven pixel on a spatial light modulator on to which the data for eachholographic pixel is written, wherein said single mathematicaltransformation transforms said digital data whilst integrally correctingsaid digital data for a finite distortion of an optical objective;writing corrected data to a spatial light modulator, wherein saidcorrected data is described by said tensor ^(μν)T_(αβ); directing saidlaser beam on to said spatial light modulator so that at least a portionof said laser beam is spatially modulated by said spatial lightmodulator to form a spatially modulated laser beam; passing saidspatially modulated laser beam through an optical objective having afinite distortion, said optical objective focusing said spatiallymodulated laser beam on to a photosensitive substrate; directing areference recording beam on to said photosensitive substrate; andforming a single-parallax composite hologram on said photosensitivesubstrate.
 43. A method as claimed in claim 42, wherein said digitaldata is acquired from a real object and comprises a plurality ofperspective views of the required hologram image.
 44. A method asclaimed in claim 42, wherein said digital data is acquired from avirtual object and comprises a plurality of perspective views of therequired hologram image.
 45. A method as claimed in claim 42, whereinsaid composite 1-step hologram is selected from the group consisting of:(i) a transmission hologram; and (ii) a reflection hologram.
 46. Amethod as claimed in claim 42, wherein said single mathematicaltransformation generates a rectangular viewing window located in frontof said hologram.
 47. A method as claimed in claim 46, wherein saidspatial light modulator is either: (i) static whilst writing saidhologram; or (ii) moving whilst writing said hologram.
 48. A method asclaimed in claim 46, wherein said viewing window is either: (i) ofsubstantially similar size to said composite hologram; or (ii) ofdifferent size to said composite hologram.
 49. A method as claimed inclaim 46, wherein said viewing window is either: (i) symmetricallylocated in front of said composite hologram; or (ii) generally offsetfrom the centre of said composite hologram.
 50. A method as claimed inclaim 46, wherein either: (i) said viewing window is located at the sameperpendicular distance from a given point in the holographic image asthe camera plane is located from the corresponding point on the objectfrom which said digital data is acquired; or (ii) said viewing window islocated at a certain perpendicular distance from a given point in theholographic image and the camera plane is located at a substantiallydifferent perpendicular distance from the corresponding point on theobject from which said digital data is acquired.
 51. A method as claimedin claim 42, wherein said digital data is generated by a real or virtualcamera which generates either: (i) a plurality of apodized images whichare centred in a frame which corresponds with the object which is to bereproduced by said hologram; or (ii) a plurality of non-apodized imageshaving frames which correspond with the object which is to be reproducedby said hologram, said frames being generally off-centred.
 52. A methodas claimed in claim 42, wherein said single mathematical transformationgenerates a scrolling viewing window located in front of said hologram.53. A method as claimed in claim 42, wherein said single mathematicaltransformation generates a viewing window having a fixed size in thehorizontal dimension and which scrolls in the vertical dimension, saidviewing window being located in front of the hologram.
 54. A method asclaimed in claim 42, wherein said single mathematical transformationgenerates a viewing window having a fixed size in the vertical dimensionand which scrolls in the horizontal dimension, said viewing window beinglocated in front of the hologram.
 55. A method as claimed in claim 52,wherein either: (i) the camera plane is located at a certain distancefrom a point on the object and the viewing plane is located atsubstantially the same distance from a corresponding point in theholographic image; or (ii) the camera plane is located at a certaindistance from a point on the object and the viewing plane is located atsubstantially a different distance from a corresponding point in theholographic image.
 56. A method as claimed of claim 42, wherein saidcomposite hologram is formed using a variable angle reference recordingbeam.
 57. A method as claimed in claim 56, wherein said composite 1-stephologram is replayed using a point-source light.
 58. A method as claimedin claim 56, wherein said composite 1-step hologram is replayed usingcollimated light.
 59. A method as claimed in claim 42, wherein saidcomposite hologram is formed using a fixed angle or collimated referencerecording beam.
 60. A method as claimed in claim 42, wherein said singlemathematical transformation additionally integrally corrects saiddigital data for the image distortion caused by the altitudinal andazimuthal reference beam angle(s) used to replay each holographic pixelof said hologram being different from the altitudinal and azimuthalreference beam angle(s) used to write each holographic pixel of saidhologram.
 61. A method as claimed in claim 42, wherein said singlemathematical transformation additionally integrally pre-distorts saiddigital data so that the data written on to said spatial light modulatoris distorted.
 62. A method as claimed in claim 61, further comprisingovercorrecting the reference recording beam using either an astigmaticor a non-astigmatic geometry.
 63. A method as claimed in claim 62,wherein the overlap of viewing windows of a plurality of holographicpixels is arranged to be maximised.
 64. A method as claimed in claim 63,wherein either: (i) the angular resolution within the overall viewingwindow of said hologram averaged over said overall viewing window ismaximised; or (ii) the angular resolution at the periphery of saidoverall viewing window is maximised.
 65. A method as claimed in claim62, wherein the overlap of viewing windows of two diagonally opposedholographic pixels is maximised.
 66. A method as claimed in claim 62,wherein the pre-distortion of said digital data and the step ofovercorrecting the reference recording beam are arranged such thatchromatic discoloration is minimized.
 67. A method as claimed in claim61, further comprising determining individual altitudinal and azimuthalreference recording angles for at least a majority of the holographicpixels forming said hologram.
 68. A method as claimed in claim 42,wherein said single mathematical transformation additionally integrallycorrects said digital data for the distortion caused by emulsionswelling of said substrate.
 69. A method as claimed in claim 42, whereinsaid single mathematical transformation additionally integrally correctssaid digital data for the distortion caused by the wavelength of lightused to replay said hologram being different from the wavelength oflight used to write said hologram.
 70. A method as claimed in claim 42,further comprising a plurality of colour channels.
 71. A method asclaimed in claim 70, further comprising a red and/or green and/or bluecolour channel.
 72. A method as claimed in claim 71, wherein a spatiallight modulator is provided for each colour channel.
 73. A method asclaimed in claim 70, wherein said composite 1-step hologram is amultiple colour hologram.
 74. A method as claimed in claim 73, whereinsaid multiple colour hologram is formed using reference recording beamshaving a first geometry and said hologram is replayed with light rayshaving a geometry different to said first geometry.
 75. A method asclaimed in claim 74, further comprising calculating the replaywavelength as a function of altitudinal and azimuthal angles for atleast a majority of the holographic pixels forming said hologram.
 76. Amethod as claimed in claim 70, further comprising calculating linearchromatic coupling tensors for each colour channel.
 77. A method asclaimed in claim 76, wherein a separate tensor ^(μν)T_(αβ) is calculatedfor each colour channel.
 78. A method as claimed in claim 77, wherein acorrected tensor is calculated for each colour channel as a linearcombination of each uncorrected component colour tensor ^(μν)T_(αβ) eachoperated on by a said chromatic coupling tensor.
 79. A method as claimedin claim 78, wherein for each holographic pixel each said correctedtensor is written to a separate spatial light modulator in such a way asto create a fully colour-corrected composite colour hologram.
 80. Amethod as claimed in claim 42, wherein said single mathematicaltransformation between the tensors ^(k)I_(ij) and ^(μν)T_(αβ) consistsof a reordering of the elements according to a set of single index lawsof the formk=ƒ₁(α, β, μ, ν, P_(l), Q_(l), H_(l), λ), i=ƒ₂(α, β, μ, ν, P_(l), Q_(l),H_(l), λ) and j=ƒ₃(α, β, μ, ν, P_(l), Q_(l), H_(l), λ), wherein thefunctions ƒ_(n) are general functions of the indicated indices, P_(l)are a set of parameters characterizing the physical characteristics ofthe hologram, Q_(l) are a set of parameters characterizing the opticalproperties of the hologram writing mechanism, H_(l) is a set ofparameters characterizing the geometrical properties of the referencerecording and reference replay beams and λ is the wavelength at whichthe hologram is recorded.